Ljubiša D. R. KočinacpersName, Farkhod G. MukhamadievpersName, Anvar K. SadullaevpersName (2022date). “Some Cardinal and Geometric Properties of the Space of Permutation Degree.” Axiomstitle·j.
Object of study — hereditary cardinal invariantsterm and geometric properties of the space of permutation degreeterm of a topological space, in particular of the Hattori spaceterm
Research question — how cardinal invariantsterm such as spreadterm, extentterm, densityterm and Lindelof number behave in the permutation-degree space, and whether the permutation-degree functor preserves homotopyterm and retractionterm
Contribution — relations among cardinal invariantsterm in the permutation-degree space of the Hattori spaceterm, and a proof that the functor of G-permutation degreeterm preserves homotopyterm and retractionterm and is a covariant homotopy functorterm
phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | the space of permutation degree of the Hattori space carries a Sorgenfrey-type topology and contains a closed discrete subset of the cardinality of the continuum — “The space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subset of cardinality 𝑐 .” | |
| claim-type | existence — “This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subset of cardinality 𝑐 . Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalent. As a result, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor.” | |
| method — evidence | — “Now we study some geometric properties of the space SP 𝐺 𝑛 𝑋 of permutation degree. In particular, we show the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. In fact, we prove that if spaces X and Y are homotopically equivalent, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalent, and conclude that the functor SP 𝐺 𝑛 is a covariant homotopy functor.” | |
| object of study — evidence | — “The current paper is devoted to the investigation of hereditary cardinal invariants (such as the hereditary density, the hereditary weak density, and the hereditary Lindelöf number) in the space of permutation degree. Additionally, the relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degree of Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) is studied. Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. As a consequence, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor. Our research complements and extends existing results in the fields of cardinal invariants and the theory of covariant functors.” | |
| research question — evidence | — “The current paper is devoted to the investigation of hereditary cardinal invariants (such as the hereditary density, the hereditary weak density, and the hereditary Lindelöf number) in the space of permutation degree. Additionally, the relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degree of Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) is studied. Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. As a consequence, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor. Our research complements and extends existing results in the fields of cardinal invariants and the theory of covariant functors.” | |
| contribution — evidence | — “This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subset of cardinality 𝑐 . Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalent. As a result, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor.” | |
| mechanism | functor lifts maps The functor of G-permutation degree sends a map f to the map carrying an orbit of an n-tuple to the orbit of the tuple of images, lifting maps to the permutation-degree space. — “For every mapping 𝑓 : 𝑋 → 𝑌 , the mapping SP 𝐺 𝑛 𝑓 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝑌 is defined [17] by the formula SP 𝐺 𝑛 𝑓 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 = [ ( 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) , … , 𝑓 ( 𝑥 𝑛 ) ) ] 𝐺 .” | mechanisms[0] |
| mechanism | homotopy lifts through the functor A homotopy between two maps lifts to a homotopy between their images under the functor, applying the homotopy coordinatewise on representatives. — “It is clear that since the mapping F is continuous, the mapping SP 𝐺 𝑛 𝐹 is also continuous. Now we will show that the mapping SP 𝐺 𝑛 𝐹 is a homotopy between the mappings SP 𝐺 𝑛 𝑓 and SP 𝐺 𝑛 𝑔 . Indeed, SP 𝐺 𝑛 𝐹 ( [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 , 0 ) = [ ( 𝐹 ( 𝑥 1 , 0 ) , 𝐹 ( 𝑥 2 , 0 ) , … , 𝐹 ( 𝑥 𝑛 , 0 ) ) ] 𝐺 = [ ( 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) , … , 𝑓 ( 𝑥 𝑛 ) ) ] 𝐺 = SP 𝐺 𝑛 𝑓 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 ; and SP 𝐺 𝑛 𝐹 ( [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 , 1 ) = [ ( 𝐹 ( 𝑥 1 , 1 ) , 𝐹 ( 𝑥 2 , 1 ) , … , 𝐹 ( 𝑥 𝑛 , 1 ) ) ] 𝐺 = [ ( 𝑔 ( 𝑥 1 ) , 𝑔 ( 𝑥 2 ) , … , 𝑔 ( 𝑥 𝑛 ) ) ] 𝐺 = SP 𝐺 𝑛 𝑔 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 .” | mechanisms[1] |
| mechanism | hereditary invariant by subspaces The hereditary version of a cardinal function is the supremum of that function over all subspaces. — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | mechanisms[2] |
| extends | existing results on cardinal invariants under covariant functors — “The current paper is devoted to the investigation of hereditary cardinal invariants (such as the hereditary density, the hereditary weak density, and the hereditary Lindelöf number) in the space of permutation degree. Additionally, the relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degree of Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) is studied. Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. As a consequence, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor. Our research complements and extends existing results in the fields of cardinal invariants and the theory of covariant functors.” | claim_relations[0] |
| builds-on | the theory of the symmetric-product / permutation-degree functor — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | claim_relations[1] |
| builds-on | prior results on homotopy properties of topological spaces — “In [14], some propositions about homotopy properties of the topological spaces were given. For instance, it was proved that contractibility, connectedness, and pathwise connectedness are homotopy property of the spaces.” | claim_relations[2] |
| sense | cardinal invariant - topology sense: a cardinal-valued function invariant under homeomorphism, describing a space — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | senses[0] |
| sense | extent versus spread - cardinal-function senses: closed-discrete versus discrete subspace cardinalities — “The extent of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularity of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf number 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .” | senses[1] |
| sense | functor - covariant homotopy sense: preserves identities, compositions and homotopies — “By a covariant homotopy functor [17], we mean an operator 𝜙 which assigns to each topological space X a space 𝜙 ( 𝑋 ) , and to each continuous mapping 𝑓 : 𝑋 → 𝑌 , a mapping 𝜙 ( 𝑓 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) satisfying the following three conditions: (i) 𝜙 preserves the identity mapping; that is, if f is the identity mapping of X, then 𝜙 ( 𝑓 ) is the identity mapping of 𝜙 ( 𝑋 ) . (ii) 𝜙 preserves compositions; that is, if 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑍 are continuous mappings, then 𝜙 ( 𝑔 ∘ 𝑓 ) = 𝜙 ( 𝑔 ) ∘ 𝜙 ( 𝑓 ) . (iii) 𝜙 preserves homotopy; that is, if a mapping 𝐹 ( 𝑥 , 𝑡 ) is a homotopy between the continuous mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 , then 𝜙 ( 𝐹 ( 𝑥 , 𝑡 ) ) is a homotopy between the mappings 𝜙 ( 𝑓 ) , 𝜙 ( 𝑔 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) .” | senses[2] |
| sense | equivalent - homeomorphic versus homotopically equivalent sense: two strengths of topological sameness — “A continuous mapping 𝑓 : 𝑋 → 𝑌 is said to be a homotopy equivalence [18] if there exists a continuous mapping 𝑔 : 𝑌 → 𝑋 such that the compositions 𝑔 ∘ 𝑓 and 𝑓 ∘ 𝑔 are homotopic to the identity mappings on X and Y, respectively. Two topological spaces X and Y are said to be homotopically equivalent (notation 𝑋 ≃ 𝑌 ) if there exists a homotopy equivalence 𝑓 : 𝑋 → 𝑌 [18].” | senses[3] |
| note | existing results on cardinal invariants under covariant functors — The paper states it complements and extends existing results in cardinal invariants and the theory of covariant functors. — “The current paper is devoted to the investigation of hereditary cardinal invariants (such as the hereditary density, the hereditary weak density, and the hereditary Lindelöf number) in the space of permutation degree. Additionally, the relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degree of Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) is studied. Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. As a consequence, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor. Our research complements and extends existing results in the fields of cardinal invariants and the theory of covariant functors.” | claim_relations[0] |
| weight | existing results on cardinal invariants under covariant functors — 0.9 — “The current paper is devoted to the investigation of hereditary cardinal invariants (such as the hereditary density, the hereditary weak density, and the hereditary Lindelöf number) in the space of permutation degree. Additionally, the relation between the spread and the extent of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degree of Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) is studied. Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. As a consequence, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functor. Our research complements and extends existing results in the fields of cardinal invariants and the theory of covariant functors.” | claim_relations[0] |
| note | the theory of the symmetric-product / permutation-degree functor — The functor of G-permutation degree and its basic properties are taken from prior work. — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | claim_relations[1] |
| weight | the theory of the symmetric-product / permutation-degree functor — 0.92 — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | claim_relations[1] |
| note | prior results on homotopy properties of topological spaces — Uses the earlier result that contractibility, connectedness and pathwise connectedness are homotopy properties. — “In [14], some propositions about homotopy properties of the topological spaces were given. For instance, it was proved that contractibility, connectedness, and pathwise connectedness are homotopy property of the spaces.” | claim_relations[2] |
| weight | prior results on homotopy properties of topological spaces — 0.9 — “In [14], some propositions about homotopy properties of the topological spaces were given. For instance, it was proved that contractibility, connectedness, and pathwise connectedness are homotopy property of the spaces.” | claim_relations[2] |
| audit | G-permutation degree —is-a→ covariant functor — Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span calls the FUNCTOR SP_G^n covariant; 'G-permutation degree of X' names the quotient space, so is-a is loose. — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | concept_edges[0] |
| note | G-permutation degree —is-a→ covariant functor — Flat categorical membership; kind cue 'is a covariant functor'. — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | concept_edges[0] |
| relation | G-permutation degree —is-a→ covariant functor — The functor of G-permutation degree is a covariant functor in the category of compact spaces. — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | concept_edges[0] |
| audit | Hattori space —distinct-from→ Sorgenfrey topology — Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span orders the topologies (finer than Euclidean, weaker than Sorgenfrey); distinct-from is loose. — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | concept_edges[2] |
| note | Hattori space —distinct-from→ Sorgenfrey topology — Differentiator stated by comparison; cue 'weaker than'. — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | concept_edges[2] |
| relation | Hattori space —distinct-from→ Sorgenfrey topology — The Hattori-space topology is finer than the Euclidean topology and weaker than the Sorgenfrey topology. — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | concept_edges[2] |
| weight | Hattori space —distinct-from→ Sorgenfrey topology — 0.85 — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | concept_edges[2] |
| relation | homotopy equivalence → cardinal invariant — Cardinal invariants agree on homeomorphic spaces, a stricter relation than homotopy equivalence. — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | unhoused_relations[0] |
| weight | homotopy equivalence → cardinal invariant — 0.85 — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | unhoused_relations[0] |
| no type for this | homotopy equivalence → cardinal invariant — A cardinal invariant is a homeomorphism invariant, not in general a homotopy invariant, so homotopy-equivalent spaces need not share it. — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | unhoused_relations[0] |
| limitation | All spaces are assumed to be completely regular, so the results are stated under that standing separation assumption. — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | limitations[0] |
| limitation | The cardinal-invariant equalities are established for the specific second permutation-degree space of the Hattori space, not for arbitrary spaces. — “In this section, we study some (hereditary) cardinal invariants (the spread, extent, density, weak density, 𝜋 -weight) of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degree of the Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) . Let us observe that the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) has a Sorgenfrey-type topology.” | limitations[1] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | The permutation-degree space of the Sorgenfrey line contains a closed discrete subset of the cardinality of the continuum. — “The space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subset of cardinality 𝑐 .” | content_tags.findings[0] |
| finding | For the Hattori space, the spread and the extent of its second permutation-degree space both equal the cardinality of the continuum. — “Let A be a subset of 𝑅 and 𝐵 ⊆ 𝑅 \ 𝐴 . If B is a closed subset of ( 𝑅 , 𝜏 ( 𝐴 ) ) which is homeomorphic to the space ( 𝑅 , 𝜏 𝑆 ) , then 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑒 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 .” | content_tags.findings[1] |
| finding | The functor of G-permutation degree preserves the homotopy and the retraction of topological spaces. — “Now we study some geometric properties of the space SP 𝐺 𝑛 𝑋 of permutation degree. In particular, we show the functor SP 𝐺 𝑛 preserves the homotopy and the retraction of topological spaces. In fact, we prove that if spaces X and Y are homotopically equivalent, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalent, and conclude that the functor SP 𝐺 𝑛 is a covariant homotopy functor.” | content_tags.findings[2] |
| finding | If two spaces are homotopically equivalent, then their images under the functor of G-permutation degree are homotopically equivalent. — “If the spaces X and Y are homotopically equivalent, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalent.” | content_tags.findings[3] |
| finding | The functor of G-permutation degree is a covariant homotopy functor. — “The functor 𝑆 𝑃 𝐺 𝑛 is a covariant homotopy functor.” | content_tags.findings[4] |
| finding | The fundamental groups of a space and its permutation-degree space are not always isomorphic. — “The fundamental groups 𝜋 1 ( 𝑋 , 𝑥 0 ) and 𝜋 1 ( 𝑆 𝑃 𝐺 𝑛 𝑋 , [ 𝑥 0 ] 𝐺 ) of the topological spaces X and 𝑆 𝑃 𝐺 𝑛 𝑋 are not always isomorphic for every 𝑥 0 ∈ 𝑋 , where [ 𝑥 0 ] 𝐺 = [ ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) ] 𝐺 = ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) .” | content_tags.findings[5] |
| definition | cardinal function — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | content_tags.definitions[0] |
| definition | hereditary cardinal invariant — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | content_tags.definitions[1] |
| definition | density — “A set 𝐴 ⊂ 𝑋 is dense in X if 𝐴 ¯ = 𝑋 . The density of X, denoted by 𝑑 ( 𝑋 ) , is defined as 𝑑 ( 𝑋 ) = min { | 𝐴 | : 𝐴 𝑖 𝑠 𝑑 𝑒 𝑛 𝑠 𝑒 𝑖 𝑛 𝑋 } [4]. A collection 𝐵 of nonempty open sets in X is said to be a π-base of X if for every nonempty open set 𝐺 ⊂ 𝑋 there is a 𝐵 ∈ 𝐵 with 𝐵 ⊂ 𝐺 . The π-weight of a space X is defined as 𝜋 𝑤 ( 𝑋 ) = min { | 𝐵 | : 𝐵 is a 𝜋 -base of 𝑋 } [1,4,6]. The weak density of a space X, denoted by 𝑤 𝑑 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that there is a 𝜋 -base 𝐵 = ∪ { 𝐵 𝛼 : 𝛼 < 𝜏 } in X, and for each 𝛼 < 𝜏 , 𝐵 𝛼 is a centered system of open sets in X [7,9,10].” | content_tags.definitions[2] |
| definition | weak density — “A set 𝐴 ⊂ 𝑋 is dense in X if 𝐴 ¯ = 𝑋 . The density of X, denoted by 𝑑 ( 𝑋 ) , is defined as 𝑑 ( 𝑋 ) = min { | 𝐴 | : 𝐴 𝑖 𝑠 𝑑 𝑒 𝑛 𝑠 𝑒 𝑖 𝑛 𝑋 } [4]. A collection 𝐵 of nonempty open sets in X is said to be a π-base of X if for every nonempty open set 𝐺 ⊂ 𝑋 there is a 𝐵 ∈ 𝐵 with 𝐵 ⊂ 𝐺 . The π-weight of a space X is defined as 𝜋 𝑤 ( 𝑋 ) = min { | 𝐵 | : 𝐵 is a 𝜋 -base of 𝑋 } [1,4,6]. The weak density of a space X, denoted by 𝑤 𝑑 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that there is a 𝜋 -base 𝐵 = ∪ { 𝐵 𝛼 : 𝛼 < 𝜏 } in X, and for each 𝛼 < 𝜏 , 𝐵 𝛼 is a centered system of open sets in X [7,9,10].” | content_tags.definitions[3] |
| definition | extent — “The extent of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularity of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf number 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .” | content_tags.definitions[4] |
| definition | Lindelöf number — “The extent of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularity of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf number 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .” | content_tags.definitions[5] |
| definition | symmetric equivalence relation — “Let X be a space. The permutation group 𝑆 𝑛 acts on the n-th power 𝑋 𝑛 of X as the permutation of coordinates: the points ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) , ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) ∈ 𝑋 𝑛 are equivalent if there exists a permutation 𝜎 ∈ 𝑆 𝑛 for which 𝑦 𝑖 = 𝑥 𝜎 ( 𝑖 ) . This equivalence relation is called the symmetric equivalence relation [17], and the set of all orbits of the action of 𝑆 𝑛 on 𝑋 𝑛 with the quotient topology is denoted by SP 𝑛 𝑋 and called the space of n-permutation degree of X [17].” | content_tags.definitions[6] |
| definition | G-permutation degree — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | content_tags.definitions[7] |
| definition | homotopy equivalence — “A continuous mapping 𝑓 : 𝑋 → 𝑌 is said to be a homotopy equivalence [18] if there exists a continuous mapping 𝑔 : 𝑌 → 𝑋 such that the compositions 𝑔 ∘ 𝑓 and 𝑓 ∘ 𝑔 are homotopic to the identity mappings on X and Y, respectively. Two topological spaces X and Y are said to be homotopically equivalent (notation 𝑋 ≃ 𝑌 ) if there exists a homotopy equivalence 𝑓 : 𝑋 → 𝑌 [18].” | content_tags.definitions[8] |
| definition | covariant homotopy functor — “By a covariant homotopy functor [17], we mean an operator 𝜙 which assigns to each topological space X a space 𝜙 ( 𝑋 ) , and to each continuous mapping 𝑓 : 𝑋 → 𝑌 , a mapping 𝜙 ( 𝑓 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) satisfying the following three conditions: (i) 𝜙 preserves the identity mapping; that is, if f is the identity mapping of X, then 𝜙 ( 𝑓 ) is the identity mapping of 𝜙 ( 𝑋 ) . (ii) 𝜙 preserves compositions; that is, if 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑍 are continuous mappings, then 𝜙 ( 𝑔 ∘ 𝑓 ) = 𝜙 ( 𝑔 ) ∘ 𝜙 ( 𝑓 ) . (iii) 𝜙 preserves homotopy; that is, if a mapping 𝐹 ( 𝑥 , 𝑡 ) is a homotopy between the continuous mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 , then 𝜙 ( 𝐹 ( 𝑥 , 𝑡 ) ) is a homotopy between the mappings 𝜙 ( 𝑓 ) , 𝜙 ( 𝑔 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) .” | content_tags.definitions[9] |
| definition | retract — “A space X which is homotopy equivalent to a point is called contractible. A subset A of a space X is a retract of X if there exists a continuous mapping 𝑟 : 𝑋 → 𝐴 , called a retraction, such that 𝑟 | 𝐴 = 1 𝐴 [18].” | content_tags.definitions[10] |
| concept | cardinal invariant — not defined in this paper's own words; a topological invariant valued in the cardinal numbers — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | concepts[0] |
| concept | cardinal function — not defined in this paper's own words; a homeomorphism-invariant assignment of an infinite cardinal to a space — “The cardinal invariants are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, density, character, Lindelöf number, tightness, and cellularity of a topological space X are some classical cardinal invariants. Many researches have been devoted to the investigation of cardinal invariants and hereditary cardinal invariants (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal function (or cardinal invariant). The hereditary version of a cardinal function 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].” | concepts[1] |
| concept | hereditary cardinal invariant — not defined in this paper's own words; the supremum of a cardinal function over subspaces — “The current paper is devoted to the investigation of hereditary cardinal invariants (such as the hereditary density, the hereditary weak density, and the hereditary Lindelöf number) in the space of permutation degree.” | concepts[2] |
| concept | density — not defined in this paper's own words; least size of a dense subset — “A set 𝐴 ⊂ 𝑋 is dense in X if 𝐴 ¯ = 𝑋 . The density of X, denoted by 𝑑 ( 𝑋 ) , is defined as 𝑑 ( 𝑋 ) = min { | 𝐴 | : 𝐴 𝑖 𝑠 𝑑 𝑒 𝑛 𝑠 𝑒 𝑖 𝑛 𝑋 } [4]. A collection 𝐵 of nonempty open sets in X is said to be a π-base of X if for every nonempty open set 𝐺 ⊂ 𝑋 there is a 𝐵 ∈ 𝐵 with 𝐵 ⊂ 𝐺 . The π-weight of a space X is defined as 𝜋 𝑤 ( 𝑋 ) = min { | 𝐵 | : 𝐵 is a 𝜋 -base of 𝑋 } [1,4,6]. The weak density of a space X, denoted by 𝑤 𝑑 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that there is a 𝜋 -base 𝐵 = ∪ { 𝐵 𝛼 : 𝛼 < 𝜏 } in X, and for each 𝛼 < 𝜏 , 𝐵 𝛼 is a centered system of open sets in X [7,9,10].” | concepts[3] |
| concept | weak density — not defined in this paper's own words; a pi-base-based weakening of density — “A set 𝐴 ⊂ 𝑋 is dense in X if 𝐴 ¯ = 𝑋 . The density of X, denoted by 𝑑 ( 𝑋 ) , is defined as 𝑑 ( 𝑋 ) = min { | 𝐴 | : 𝐴 𝑖 𝑠 𝑑 𝑒 𝑛 𝑠 𝑒 𝑖 𝑛 𝑋 } [4]. A collection 𝐵 of nonempty open sets in X is said to be a π-base of X if for every nonempty open set 𝐺 ⊂ 𝑋 there is a 𝐵 ∈ 𝐵 with 𝐵 ⊂ 𝐺 . The π-weight of a space X is defined as 𝜋 𝑤 ( 𝑋 ) = min { | 𝐵 | : 𝐵 is a 𝜋 -base of 𝑋 } [1,4,6]. The weak density of a space X, denoted by 𝑤 𝑑 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that there is a 𝜋 -base 𝐵 = ∪ { 𝐵 𝛼 : 𝛼 < 𝜏 } in X, and for each 𝛼 < 𝜏 , 𝐵 𝛼 is a centered system of open sets in X [7,9,10].” | concepts[4] |
| concept | extent — not defined in this paper's own words; supremum of sizes of closed discrete subspaces — “The extent of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularity of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf number 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .” | concepts[5] |
| concept | spread — not defined in this paper's own words; supremum of sizes of discrete subspaces — “In this section, we study some (hereditary) cardinal invariants (the spread, extent, density, weak density, 𝜋 -weight) of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degree of the Hattori space ( 𝑅 , 𝜏 ( 𝐴 ) ) . Let us observe that the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) has a Sorgenfrey-type topology.” | concepts[6] |
| concept | cellularity — not defined in this paper's own words; the Souslin number, bounding disjoint open families — “The extent of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularity of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf number 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .” | concepts[7] |
| concept | Lindelöf number — not defined in this paper's own words; least bound on subcovers of open covers — “The extent of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularity of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf number 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .” | concepts[8] |
| concept | permutation degree — not defined in this paper's own words; the quotient of a power of a space by coordinate permutations — “Let X be a space. The permutation group 𝑆 𝑛 acts on the n-th power 𝑋 𝑛 of X as the permutation of coordinates: the points ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) , ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) ∈ 𝑋 𝑛 are equivalent if there exists a permutation 𝜎 ∈ 𝑆 𝑛 for which 𝑦 𝑖 = 𝑥 𝜎 ( 𝑖 ) . This equivalence relation is called the symmetric equivalence relation [17], and the set of all orbits of the action of 𝑆 𝑛 on 𝑋 𝑛 with the quotient topology is denoted by SP 𝑛 𝑋 and called the space of n-permutation degree of X [17].” | concepts[9] |
| concept | symmetric equivalence relation — not defined in this paper's own words; equivalence of tuples up to a permutation of coordinates — “Let X be a space. The permutation group 𝑆 𝑛 acts on the n-th power 𝑋 𝑛 of X as the permutation of coordinates: the points ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) , ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) ∈ 𝑋 𝑛 are equivalent if there exists a permutation 𝜎 ∈ 𝑆 𝑛 for which 𝑦 𝑖 = 𝑥 𝜎 ( 𝑖 ) . This equivalence relation is called the symmetric equivalence relation [17], and the set of all orbits of the action of 𝑆 𝑛 on 𝑋 𝑛 with the quotient topology is denoted by SP 𝑛 𝑋 and called the space of n-permutation degree of X [17].” | concepts[10] |
| concept | G-permutation degree — not defined in this paper's own words; permutation degree under a subgroup of the symmetric group — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | concepts[11] |
| concept | covariant functor — not defined in this paper's own words; an operation on spaces and maps preserving identities and compositions — “The following generalization of the permutation degree will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degree of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functor in the category of compact spaces and is called the functor of G-permutation degree [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].” | concepts[12] |
| concept | homotopy — not defined in this paper's own words; a continuous deformation between two maps — “Continuous mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 are homotopic, denoted by 𝑓 ≃ 𝑔 , if there is a continuous mapping 𝐹 : 𝑋 × 𝐼 → 𝑌 such that 𝐹 ( 𝑥 , 0 ) = 𝑓 ( 𝑥 ) and 𝐹 ( 𝑥 , 1 ) = 𝑔 ( 𝑥 ) . F is called a homotopy between f and g [18].” | concepts[13] |
| concept | homotopy equivalence — not defined in this paper's own words; a map with a homotopy inverse — “A continuous mapping 𝑓 : 𝑋 → 𝑌 is said to be a homotopy equivalence [18] if there exists a continuous mapping 𝑔 : 𝑌 → 𝑋 such that the compositions 𝑔 ∘ 𝑓 and 𝑓 ∘ 𝑔 are homotopic to the identity mappings on X and Y, respectively. Two topological spaces X and Y are said to be homotopically equivalent (notation 𝑋 ≃ 𝑌 ) if there exists a homotopy equivalence 𝑓 : 𝑋 → 𝑌 [18].” | concepts[14] |
| concept | homotopically equivalent — not defined in this paper's own words; related by a homotopy equivalence — “A continuous mapping 𝑓 : 𝑋 → 𝑌 is said to be a homotopy equivalence [18] if there exists a continuous mapping 𝑔 : 𝑌 → 𝑋 such that the compositions 𝑔 ∘ 𝑓 and 𝑓 ∘ 𝑔 are homotopic to the identity mappings on X and Y, respectively. Two topological spaces X and Y are said to be homotopically equivalent (notation 𝑋 ≃ 𝑌 ) if there exists a homotopy equivalence 𝑓 : 𝑋 → 𝑌 [18].” | concepts[15] |
| concept | covariant homotopy functor — not defined in this paper's own words; a functor preserving identities, compositions and homotopies — “By a covariant homotopy functor [17], we mean an operator 𝜙 which assigns to each topological space X a space 𝜙 ( 𝑋 ) , and to each continuous mapping 𝑓 : 𝑋 → 𝑌 , a mapping 𝜙 ( 𝑓 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) satisfying the following three conditions: (i) 𝜙 preserves the identity mapping; that is, if f is the identity mapping of X, then 𝜙 ( 𝑓 ) is the identity mapping of 𝜙 ( 𝑋 ) . (ii) 𝜙 preserves compositions; that is, if 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑍 are continuous mappings, then 𝜙 ( 𝑔 ∘ 𝑓 ) = 𝜙 ( 𝑔 ) ∘ 𝜙 ( 𝑓 ) . (iii) 𝜙 preserves homotopy; that is, if a mapping 𝐹 ( 𝑥 , 𝑡 ) is a homotopy between the continuous mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 , then 𝜙 ( 𝐹 ( 𝑥 , 𝑡 ) ) is a homotopy between the mappings 𝜙 ( 𝑓 ) , 𝜙 ( 𝑔 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) .” | concepts[16] |
| concept | retract — not defined in this paper's own words; a subset fixed by a retraction from the whole space — “A space X which is homotopy equivalent to a point is called contractible. A subset A of a space X is a retract of X if there exists a continuous mapping 𝑟 : 𝑋 → 𝐴 , called a retraction, such that 𝑟 | 𝐴 = 1 𝐴 [18].” | concepts[17] |
| concept | retraction — not defined in this paper's own words; a map onto a subset fixing it pointwise — “A space X which is homotopy equivalent to a point is called contractible. A subset A of a space X is a retract of X if there exists a continuous mapping 𝑟 : 𝑋 → 𝐴 , called a retraction, such that 𝑟 | 𝐴 = 1 𝐴 [18].” | concepts[18] |
| concept | contractible — homotopy equivalent to a point — “A space X which is homotopy equivalent to a point is called contractible. A subset A of a space X is a retract of X if there exists a continuous mapping 𝑟 : 𝑋 → 𝐴 , called a retraction, such that 𝑟 | 𝐴 = 1 𝐴 [18].” | concepts[19] |
| concept | fundamental group — not defined in this paper's own words; the group of homotopy classes of loops based at a point — “Let X be a space and 𝑥 0 𝑖 𝑛 𝑋 . A path in X beginning and ending at 𝑥 0 is called a loop [18] based at 𝑥 0 . Denote by 𝜋 1 ( 𝑋 , 𝑥 0 ) the set of all equivalence classes [ 𝑓 ] of loops in X based at 𝑥 0 . 𝜋 1 ( 𝑋 , 𝑥 0 ) with the operation ∗ is a group, where the identity element of the group is [ 𝑒 𝑥 ] and the inverse element of [ 𝑓 ] is [ 𝑓 ¯ ( 𝑡 ) ] = [ 𝑓 ( 1 − 𝑡 ) ] . We call 𝜋 1 ( 𝑋 , 𝑥 0 ) the fundamental group [18].” | concepts[20] |
| concept | Hattori space — not defined in this paper's own words; a topology on the reals interpolating between the Euclidean and Sorgenfrey topologies — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | concepts[21] |
| concept | Sorgenfrey topology — not defined in this paper's own words; the lower-limit topology on the real line — “Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topology [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori space over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topology 𝜏 𝑆 [16].” | concepts[22] |
| concept | closed discrete subset — not defined in this paper's own words; a subspace that is both closed and discrete — “The space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subset of cardinality 𝑐 .” | concepts[23] |
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Some Cardinal and Geometric Properties of the Space of Permutation Degreeterm Abstract Abstract This paper is devoted to the investigation of cardinal invariantsterm such as the hereditary densityterm, hereditary weak densityterm, and hereditary Lindelöf numberterm. The relation between the spreadterm and the extentterm of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degreeterm of the Hattori spaceterm is discussed. In particular, it is shown that the space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subsetterm of cardinality 𝑐 . Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopyterm and the retractionterm of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalentterm, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalentterm. As a result, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functorterm. Keywords: extentterm; Lindelöf numberterm; hereditary cardinal invariantterm; homotopically equivalentterm; covariant homotopy functorterm; retractterm MSC: 18F60; 18A05; 54A25; 55P99 1. Introduction The cardinal invariantsterm are considered as topological invariants with values in the class of all cardinal numbers, and are used to describe various topological properties of spaces. For example, the weight, 𝜋 -weight, network weight, densityterm, character, Lindelöf numberterm, tightness, and cellularityterm of a topological space X are some classical cardinal invariantsterm. Many researches have been devoted to the investigation of cardinal invariantsterm and hereditary cardinal invariantsterm (see, for example, [1,2,3,4,5,6]) and their important role in topology. Recall that a function 𝜑 : Top → Card from the class Top of topological spaces to the set Card of infinite cardinals such that 𝜑 ( 𝑋 ) = 𝜑 ( 𝑌 ) whenever X and Y are homeomorphic, is called a cardinal functionterm (or cardinal invariantterm). The hereditary version of a cardinal functionterm 𝜑 , denoted ℎ 𝜑 , is defined as ℎ 𝜑 ( 𝑋 ) = sup { 𝜑 ( 𝑌 ) : 𝑌 ⊂ 𝑋 } [1,4,5,6].
In recent research, the interest in the theory of cardinal invariantsterm and their behavior under the influence of various covariant functorsterm is increasing (see, for example, [7,8,9]). In [10], the authors investigated several cardinal invariantsterm under the influence of some seminormal and normal functors. In the investigations in [11,12], the concept of symmetric product of a topological space is introduced. In particular, in [13] the functor SP 𝐺 𝑛 is studied, and some cardinal and topological properties of this functor were investigated. In [14], some propositions about homotopyterm properties of the topological spaces were proved. For instance, it was proved that, contractibility, connectedness, and pathwise connectedness are homotopyterm properties of the spaces. In our work, we prove that if the mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 are homotopic, then the mappings SP 𝐺 𝑛 𝑓 , SP 𝐺 𝑛 𝑔 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝑌 are also homotopic.
The current paper is devoted to the investigation of hereditary cardinal invariantsterm (such as the hereditary densityterm, the hereditary weak densityterm, and the hereditary Lindelöf numberterm) in the space of permutation degreeterm. Additionally, the relation between the spreadterm and the extentterm of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degreeterm of Hattori spaceterm ( 𝑅 , 𝜏 ( 𝐴 ) ) is studied. Moreover, it is shown that the functor SP 𝐺 𝑛 preserves the homotopyterm and the retractionterm of topological spaces. As a consequence, it has been proved that the functor SP 𝐺 𝑛 is a covariant homotopy functorterm. Our research complements and extends existing results in the fields of cardinal invariantsterm and the theory of covariant functorsterm.
The paper is organized as follows. In Section 2, we recall basic notions and notation that will be used in the rest of the study. In Section 3, we study hereditary cardinal invariantsterm and obtain some results for the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degreeterm of the Hattori spaceterm ( 𝑅 , 𝜏 ( 𝐴 ) ) . Finally, in Section 4, we study some geometric properties of the space SP 𝐺 𝑛 𝑋 of permutation degreeterm of a space X.
Throughout the paper, all spaces are assumed to be completely regular; 𝜏 denotes an infinite cardinal number; and by 𝜔 and 𝑐 we denote the countable cardinal number and the cardinality of continuum, respectively. The real line with the Sorgenfrey topologyterm [4] is denoted by ( 𝑅 , 𝜏 𝑆 ) ; and for 𝐴 ⊂ 𝑅 by ( 𝑅 , 𝜏 ( 𝐴 ) ) we denote the Hattori spaceterm over A. Recall that in [15], the following generalization of the Sorgengfey line was defined: if 𝐴 ⊂ 𝑅 , then 𝜏 ( 𝐴 ) denotes the topology on 𝑅 , in which each point 𝑎 ∈ 𝐴 has the usual Euclidean neighborhoods, and basic neighborhoods of a point 𝑥 ∈ 𝑅 \ 𝐴 are of the form [ 𝑥 , 𝜀 ) , 𝜀 > 0 . Notice that for 𝐴 ⊂ 𝑅 , the topology 𝜏 ( 𝐴 ) is finer than the usual Euclidean topology on 𝑅 and weaker than the Sorgenfrey topologyterm 𝜏 𝑆 [16].
2. Preliminaries For convenience of the reader, we give some notation, concepts, and statements that are widely used in this article. For a space X, the group of all permutations of X is denoted by 𝑆 ( 𝑋 ) and called the permutation group of X. If 𝑋 = { 1 , 2 … , 𝑛 } , then we write 𝑆 𝑛 instead of 𝑆 ( 𝑋 ) .
Let X be a space. The permutation group 𝑆 𝑛 acts on the n-th power 𝑋 𝑛 of X as the permutation of coordinates: the points ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) , ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) ∈ 𝑋 𝑛 are equivalent if there exists a permutation 𝜎 ∈ 𝑆 𝑛 for which 𝑦 𝑖 = 𝑥 𝜎 ( 𝑖 ) . This equivalence relation is called the symmetric equivalence relationterm [17], and the set of all orbits of the action of 𝑆 𝑛 on 𝑋 𝑛 with the quotient topology is denoted by SP 𝑛 𝑋 and called the space of n-permutation degree of X [17].
The following generalization of the permutation degreeterm will be also used in what follows. If G be a subgroup of the group 𝑆 𝑛 , then G also acts on 𝑋 𝑛 as the group of permutations of coordinates, and generates an equivalence relation called the G-symmetric equivalence relation [17]. The quotient space of 𝑋 𝑛 under this relation is called G-permutation degreeterm of X and is denoted by SP 𝐺 𝑛 𝑋 , and the quotient mapping from 𝑋 𝑛 to SP 𝐺 𝑛 𝑋 is denoted by 𝛽 𝑛 , 𝐺 𝑠 . Observe that SP 𝐺 𝑛 is a covariant functorterm in the category of compact spaces and is called the functor of G-permutation degreeterm [17]. Clearly, if 𝐺 = 𝑆 𝑛 , then SP 𝐺 𝑛 = SP 𝑛 , and if G contains only the identity element, SP 𝐺 𝑛 𝑋 = 𝑋 𝑛 [17].
In [12], it is proved that the quotient mapping 𝛽 𝑛 , 𝐺 𝑠 : 𝑋 𝑛 → SP 𝐺 𝑛 𝑋 is continuous, open, and closed surjection.
For every mapping 𝑓 : 𝑋 → 𝑌 , the mapping SP 𝐺 𝑛 𝑓 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝑌 is defined [17] by the formula SP 𝐺 𝑛 𝑓 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 = [ ( 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) , … , 𝑓 ( 𝑥 𝑛 ) ) ] 𝐺 .
A set 𝐴 ⊂ 𝑋 is dense in X if 𝐴 ¯ = 𝑋 . The densityterm of X, denoted by 𝑑 ( 𝑋 ) , is defined as 𝑑 ( 𝑋 ) = min { | 𝐴 | : 𝐴 𝑖 𝑠 𝑑 𝑒 𝑛 𝑠 𝑒 𝑖 𝑛 𝑋 } [4]. A collection 𝐵 of nonempty open sets in X is said to be a π-base of X if for every nonempty open set 𝐺 ⊂ 𝑋 there is a 𝐵 ∈ 𝐵 with 𝐵 ⊂ 𝐺 . The π-weight of a space X is defined as 𝜋 𝑤 ( 𝑋 ) = min { | 𝐵 | : 𝐵 is a 𝜋 -base of 𝑋 } [1,4,6]. The weak densityterm of a space X, denoted by 𝑤 𝑑 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that there is a 𝜋 -base 𝐵 = ∪ { 𝐵 𝛼 : 𝛼 < 𝜏 } in X, and for each 𝛼 < 𝜏 , 𝐵 𝛼 is a centered system of open sets in X [7,9,10].
For definitions of the following cardinal functionsterm, see [1,4,5,6].
The extentterm of a space X, denoted by 𝑒 ( 𝑋 ) , is defined as 𝑒 ( 𝑋 ) = sup { | 𝑌 | : 𝑌 is a closed discrete subspace in 𝑋 } . The Souslin number or cellularityterm of the space X, denoted by 𝑐 ( 𝑋 ) , is the smallest cardinal number 𝜏 ≥ 𝜔 such that every family or pairwise, disjoint, non-empty open subset of X has cardinality ≤ 𝜏 . The Lindelöf numberterm 𝑙 ( 𝑋 ) of X is the smallest cardinal number 𝜏 such that each open cover of X has a subcover of cardinality ≤ 𝜏 .
A continuous mapping 𝑓 : [ 0 , 1 ] → 𝑋 is called a path in X. 𝑓 ( 0 ) is called the initial point, and 𝑓 ( 1 ) the final point of this path. If 𝑥 ∈ 𝑋 , then the constant path 𝑒 𝑥 : 𝐼 → 𝑋 is defined by 𝑒 𝑥 ( 𝑡 ) = 𝑥 for all 𝑡 ∈ 𝐼 . A space X is path connected if for any two points 𝑥 0 , 𝑥 1 ∈ 𝑋 there is a path from 𝑥 0 to 𝑥 1 [18].
Continuous mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 are homotopic, denoted by 𝑓 ≃ 𝑔 , if there is a continuous mapping 𝐹 : 𝑋 × 𝐼 → 𝑌 such that 𝐹 ( 𝑥 , 0 ) = 𝑓 ( 𝑥 ) and 𝐹 ( 𝑥 , 1 ) = 𝑔 ( 𝑥 ) . F is called a homotopyterm between f and g [18].
Consider the mappings 𝑓 ( 𝑥 ) = ( cos ( 𝜋 𝑥 ) , sin ( 𝜋 𝑥 ) ) and 𝑔 ( 𝑥 ) = ( cos ( 𝜋 𝑥 ) , − sin ( 𝜋 𝑥 ) ) . These mappings are homotopic. We can define the homotopyterm 𝐹 : 𝐼 × 𝐼 → 𝑅 2 between f and g as follows: 𝐹 ( 𝑥 , 𝑡 ) = ( cos ( 𝜋 𝑥 ) , ( 1 − 2 𝑡 ) sin ( 𝜋 𝑥 ) ) . Indeed, F is continuous and 𝐹 ( 𝑥 , 0 ) = ( cos ( 𝜋 𝑥 ) , sin ( 𝜋 𝑥 ) ) = 𝑓 ( 𝑥 ) , 𝐹 ( 𝑥 , 1 ) = ( cos ( 𝜋 𝑥 ) , − sin ( 𝜋 𝑥 ) ) = 𝑔 ( 𝑥 ) (see [18]).
A continuous mapping 𝑓 : 𝑋 → 𝑌 is said to be a homotopy equivalenceterm [18] if there exists a continuous mapping 𝑔 : 𝑌 → 𝑋 such that the compositions 𝑔 ∘ 𝑓 and 𝑓 ∘ 𝑔 are homotopic to the identity mappings on X and Y, respectively. Two topological spaces X and Y are said to be homotopically equivalentterm (notation 𝑋 ≃ 𝑌 ) if there exists a homotopy equivalenceterm 𝑓 : 𝑋 → 𝑌 [18].
By a covariant homotopy functorterm [17], we mean an operator 𝜙 which assigns to each topological space X a space 𝜙 ( 𝑋 ) , and to each continuous mapping 𝑓 : 𝑋 → 𝑌 , a mapping 𝜙 ( 𝑓 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) satisfying the following three conditions: (i) 𝜙 preserves the identity mapping; that is, if f is the identity mapping of X, then 𝜙 ( 𝑓 ) is the identity mapping of 𝜙 ( 𝑋 ) .
(ii) 𝜙 preserves compositions; that is, if 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑍 are continuous mappings, then 𝜙 ( 𝑔 ∘ 𝑓 ) = 𝜙 ( 𝑔 ) ∘ 𝜙 ( 𝑓 ) .
(iii) 𝜙 preserves homotopyterm; that is, if a mapping 𝐹 ( 𝑥 , 𝑡 ) is a homotopyterm between the continuous mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 , then 𝜙 ( 𝐹 ( 𝑥 , 𝑡 ) ) is a homotopyterm between the mappings 𝜙 ( 𝑓 ) , 𝜙 ( 𝑔 ) : 𝜙 ( 𝑋 ) → 𝜙 ( 𝑌 ) .
A space X which is homotopyterm equivalent to a point is called contractibleterm. A subset A of a space X is a retractterm of X if there exists a continuous mapping 𝑟 : 𝑋 → 𝐴 , called a retractionterm, such that 𝑟 | 𝐴 = 1 𝐴 [18].
A property P of topological spaces is called a homotopyterm property if it is preserved by all homotopy equivalencesterm. More precisely, P is a homotopyterm property if and only if for an arbitrary homotopy equivalenceterm 𝑓 : 𝑋 → 𝑌 , if X has P, then Y also has P [18].
3. Some Cardinal Properties of the Space of Permutation Degreeterm In this section, we study some (hereditary) cardinal invariantsterm (the spreadterm, extentterm, densityterm, weak densityterm, 𝜋 -weight) of the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of permutation degreeterm of the Hattori spaceterm ( 𝑅 , 𝜏 ( 𝐴 ) ) . Let us observe that the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) has a Sorgenfrey-type topology.
We begin with the following two lemmas.
The space SP 2 ( 𝑅 , 𝜏 𝑆 ) contains a closed discrete subsetterm of cardinality 𝑐 .
Note that the subset 𝑌 = { ( 𝑥 , 𝑦 ) ∈ ( 𝑅 , 𝜏 𝑆 ) 2 : 𝑥 ≥ 𝑦 } of ( 𝑅 , 𝜏 𝑆 ) 2 is homeomorphic to the space SP 2 ( 𝑅 , 𝜏 𝑆 ) , and the set 𝑍 = { ( 𝑥 , 𝑦 ) ∈ 𝑌 : 𝑦 = − 𝑥 , 𝑥 > 0 } is closed and discrete in Y and has cardinality 𝑐 . □ Let Y be a subset of a Hausdorff topological space X and 𝑍 = { 𝐹 ∈ SP 2 𝑋 : 𝐹 ⊂ 𝑌 } ⊂ SP 2 𝑋 . Then: (i) The space SP 2 𝑌 is homeomorphic to the subspace Z of the space SP 2 𝑋 ; (ii) The set Z is open in SP 2 𝑋 whenever Y is open in X; (iii) The set Z is closed in SP 2 𝑋 whenever Y is closed in X; (iv) The set Z is clopen in SP 2 𝑋 whenever Y is clopen in X.
(i) It is known [17] and easy to check that the space exp 2 𝑌 is homeomorphic to the subspace Z of the space exp 2 𝑋 . In [17], it is shown that the space SP 2 𝑌 is homeomorphic to the space exp 2 𝑌 . Hence, we have that the space SP 2 𝑌 is homeomorphic to 𝑍 ⊂ SP 2 𝑋 .
(ii), (iii), and (iv) follow from the fact that the set Z is open (closed, clopen) in exp 2 𝑋 whenever Y is open (closed, clopen) in X [17] and the mentioned result that SP 2 𝑋 is homeomorphic to exp 2 𝑋 . □ From Lemmas 1 and 2 we have: Let A be a subset of 𝑅 and 𝐵 ⊆ 𝑅 \ 𝐴 . If B is a (closed) subset of ( 𝑅 , 𝜏 ( 𝐴 ) ) which is homeomorphic to the space ( 𝑅 , 𝜏 𝑆 ) , then the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) contains a (closed) discrete subset of cardinality 𝑐 .
In [16], it was shown that if 𝐴 ⊆ 𝑅 and 𝐵 ⊆ 𝑅 \ 𝐴 , then 𝜏 ( 𝐴 ) | 𝐵 = 𝜏 𝑆 | 𝐵 . Recall that the set ( 𝑅 , 𝜏 ( 𝐴 ) ) is a closed subset of SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) . Hence, each (closed) discrete subset M of ( 𝑅 , 𝜏 ( 𝐴 ) ) with cardinality 𝑐 (which exists by Lemma 1) is a (closed) discrete subset of SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) with cardinality 𝑐 . □ Let A be a subset of 𝑅 , and Y be a subspace of ( 𝑅 , 𝜏 ( 𝐴 ) ) . Then, | 𝜏 ( 𝐴 ) 𝑌 | ≤ 𝑐 . In addition, | SP 2 𝑌 | ≤ 𝑐 .
It is enough to show that | 𝜏 ( 𝐴 ) | ≤ 𝑐 . Let 𝐵 be a base for ( 𝑅 , 𝜏 ( 𝐴 ) ) of cardinality ≤ 𝑐 . Since the space ( 𝑅 , 𝜏 ( 𝐴 ) ) is hereditary Lindelöf, each open subset of ( 𝑅 , 𝜏 ( 𝐴 ) ) is an union of countably many elements of 𝐵 . Hence, | 𝜏 ( 𝐴 ) | ≤ 𝑐 𝜔 = ( 2 𝜔 ) 𝜔 = 2 𝜔 = 𝑐 . □ Let A be a subset of 𝑅 and 𝐵 ⊆ 𝑅 \ 𝐴 . If B is a closed subset of ( 𝑅 , 𝜏 ( 𝐴 ) ) which is homeomorphic to the space ( 𝑅 , 𝜏 𝑆 ) , then 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑒 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 .
By Proposition 1 we have 𝑒 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≥ 𝑐 . Note that 𝑒 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≤ 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≤ | ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) | .
It follows from Proposition 2 that | ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) | ≤ | 𝑒 𝑥 𝑝 ( 𝑅 , 𝜏 ( 𝐴 ) ) | ≤ 𝑐 .
Thus, we get 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑒 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 .
Let A be a subset of 𝑅 and 𝐵 ⊆ 𝑅 \ 𝐴 . If B is a subset of ( 𝑅 , 𝜏 ( 𝐴 ) ) which is homeomorphic to the space ( 𝑅 , 𝜏 𝑆 ) , then 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 .
By Proposition 1 we have 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≥ 𝑐 . Note that 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≤ | ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) | . It follows from Proposition 2 that | ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) | ≤ 𝑐 . Thus, we get 𝑠 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 . □ Let A be a subset of 𝑅 and 𝜑 ∈ { 𝑑 , 𝑒 , 𝑐 } (resp. 𝜑 ∈ { 𝑤 𝑑 , 𝑙 , 𝜋 𝑤 } ). If B is a subset of ( 𝑅 , 𝜏 ( 𝐴 ) ) which is homeomorphic to the space ( 𝑅 , 𝜏 𝑆 ) , then ℎ 𝜑 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 (viz., ℎ 𝜑 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≥ 𝑐 ).
In fact, note that ℎ 𝑒 = 𝑠 and ℎ 𝑐 = 𝑠 . Since ℎ 𝑐 ( SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) ≥ 𝑐 (by Proposition 1), we have the equalities. The inequalities also trivially follow from Proposition 1. □ Let 𝐴 ⊆ 𝑅 . Note that the family 𝐵 = { ( 𝑟 1 , 𝑟 2 ) : 𝑟 1 , 𝑟 2 ∈ 𝑄 , 𝑟 1 < 𝑟 2 } is a π-base for the space ( 𝑅 , 𝜏 ( 𝐴 ) ) and 𝐵 𝑛 = { ∏ 𝑖 = 1 𝑛 𝐵 𝑖 : 𝐵 𝑖 ∈ 𝐵 } is also a π-base for the space ( 𝑅 , 𝜏 ( 𝐴 ) ) 𝑛 . Hence, 𝜋 𝑤 ( 𝑅 , 𝜏 ( 𝐴 ) ) 𝑛 = 𝑐 ( 𝑅 , 𝜏 ( 𝐴 ) ) 𝑛 = 𝜔 . Similarly, the family SP 𝑛 𝐵 𝑛 = { SP 𝑛 𝐵 𝑖 = 𝛽 𝑛 𝑠 ( 𝐵 𝑖 ) : 𝐵 𝑖 ∈ 𝐵 𝑛 } is a π-base for the space SP 𝑛 ( 𝑅 , 𝜏 ( 𝐴 ) ) . This shows that 𝜋 𝑤 ( SP 𝑛 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝑐 ( SP 𝑛 ( 𝑅 , 𝜏 ( 𝐴 ) ) ) = 𝜔 .
4. Some Geometric Properties of the Space of Permutation Degreeterm Now we study some geometric properties of the space SP 𝐺 𝑛 𝑋 of permutation degreeterm. In particular, we show the functor SP 𝐺 𝑛 preserves the homotopyterm and the retractionterm of topological spaces. In fact, we prove that if spaces X and Y are homotopically equivalentterm, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalentterm, and conclude that the functor SP 𝐺 𝑛 is a covariant homotopy functorterm.
If the mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 are homotopic, then the mappings SP 𝐺 𝑛 𝑓 , SP 𝐺 𝑛 𝑔 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝑌 are also homotopic.
Assume that the mappings 𝑓 , 𝑔 : 𝑋 → 𝑌 are homotopic. Then there exists a continuous mapping 𝐹 : 𝑋 × 𝐼 → 𝑌 such that 𝐹 ( 𝑥 , 0 ) = 𝑓 ( 𝑥 ) and 𝐹 ( 𝑥 , 1 ) = 𝑔 ( 𝑥 ) . On the other hand, we have SP 𝐺 𝑛 𝑓 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 = [ ( 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) , … , 𝑓 ( 𝑥 𝑛 ) ) ] 𝐺 , SP 𝐺 𝑛 𝑔 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 = [ ( 𝑔 ( 𝑥 1 ) , 𝑔 ( 𝑥 2 ) , … , 𝑔 ( 𝑥 𝑛 ) ) ] 𝐺 .
Now we define the mapping SP 𝐺 𝑛 𝐹 ( [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 , 𝑡 ) = [ ( 𝐹 ( 𝑥 1 , 𝑡 ) , 𝐹 ( 𝑥 2 , 𝑡 ) , … , 𝐹 ( 𝑥 𝑛 , 𝑡 ) ) ] 𝐺 .
It is clear that since the mapping F is continuous, the mapping SP 𝐺 𝑛 𝐹 is also continuous. Now we will show that the mapping SP 𝐺 𝑛 𝐹 is a homotopyterm between the mappings SP 𝐺 𝑛 𝑓 and SP 𝐺 𝑛 𝑔 . Indeed, SP 𝐺 𝑛 𝐹 ( [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 , 0 ) = [ ( 𝐹 ( 𝑥 1 , 0 ) , 𝐹 ( 𝑥 2 , 0 ) , … , 𝐹 ( 𝑥 𝑛 , 0 ) ) ] 𝐺 = [ ( 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) , … , 𝑓 ( 𝑥 𝑛 ) ) ] 𝐺 = SP 𝐺 𝑛 𝑓 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 ; and SP 𝐺 𝑛 𝐹 ( [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 , 1 ) = [ ( 𝐹 ( 𝑥 1 , 1 ) , 𝐹 ( 𝑥 2 , 1 ) , … , 𝐹 ( 𝑥 𝑛 , 1 ) ) ] 𝐺 = [ ( 𝑔 ( 𝑥 1 ) , 𝑔 ( 𝑥 2 ) , … , 𝑔 ( 𝑥 𝑛 ) ) ] 𝐺 = SP 𝐺 𝑛 𝑔 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 .
This means that SP 𝐺 𝑛 𝑓 ≃ SP 𝐺 𝑛 𝑔 . □ If the spaces X and Y are homotopically equivalentterm, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are also homotopically equivalentterm.
Suppose that the spaces X and Y are homotopically equivalentterm. Then there exist two continuous mappings 𝑓 : 𝑋 → 𝑌 and 𝑔 : 𝑌 → 𝑋 such that 𝑓 ∘ 𝑔 ≃ 𝑖 𝑑 𝑌 and 𝑔 ∘ 𝑓 ≃ 𝑖 𝑑 𝑋 . This means that there are two homotopyterm 𝐹 ( 𝑦 , 𝑡 ) and 𝐻 ( 𝑥 , 𝑡 ) such that 𝐹 ( 𝑦 , 0 ) = ( 𝑓 ∘ 𝑔 ) ( 𝑦 ) , 𝐹 ( 𝑦 , 1 ) = 𝑦 𝑎 𝑛 𝑑 ( 𝑥 , 0 ) = ( 𝑔 ∘ 𝑓 ) ( 𝑥 ) , 𝐻 ( 𝑥 , 1 ) = 𝑥 .
Consider the compositions SP 𝐺 𝑛 𝑓 ∘ SP 𝐺 𝑛 𝑔 : SP 𝐺 𝑛 𝑌 → SP 𝐺 𝑛 𝑌 and SP 𝐺 𝑛 𝑔 ∘ SP 𝐺 𝑛 𝑓 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝑋 of the mappings SP 𝐺 𝑛 𝑓 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝑌 and SP 𝐺 𝑛 𝑔 : SP 𝐺 𝑛 𝑌 → SP 𝐺 𝑛 𝑋 defined by 𝑆 𝑃 𝐺 𝑛 𝑓 ∘ 𝑆 𝑃 𝐺 𝑛 𝑔 ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) 𝐺 = ( 𝑓 ∘ 𝑔 ) ( 𝑦 1 ) , ( 𝑓 ∘ 𝑔 ) ( 𝑦 2 ) , … , ( 𝑓 ∘ 𝑔 ) ( 𝑦 𝑛 ) 𝐺 and 𝑆 𝑃 𝐺 𝑛 𝑔 ∘ 𝑆 𝑃 𝐺 𝑛 𝑓 ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) 𝐺 = ( 𝑓 ∘ 𝑔 ) ( 𝑥 1 ) , ( 𝑓 ∘ 𝑔 ) ( 𝑥 2 ) , … , ( 𝑓 ∘ 𝑔 ) ( 𝑥 𝑛 ) 𝐺 .
One can easily check that the mapping 𝑆 𝑃 𝐺 𝑛 𝐹 [ ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑛 ) ] 𝐺 , 𝑡 = ( 𝐹 ( 𝑦 1 , 𝑡 ) , 𝐹 ( 𝑦 2 , 𝑡 ) , … , 𝐹 ( 𝑦 𝑛 , 𝑡 ) ) 𝐺 is a homotopyterm between 𝑆 𝑃 𝐺 𝑛 𝑓 ∘ 𝑆 𝑃 𝐺 𝑛 𝑔 and 𝑖 𝑑 𝑆 𝑃 𝐺 𝑛 𝑌 .
Similarly, 𝑆 𝑃 𝐺 𝑛 𝐻 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 , 𝑡 = ( 𝐻 ( 𝑥 1 , 𝑡 ) , 𝐻 ( 𝑥 2 , 𝑡 ) , … , 𝐻 ( 𝑥 𝑛 , 𝑡 ) ) 𝐺 is a homotopyterm between 𝑆 𝑃 𝐺 𝑛 𝑔 ∘ 𝑆 𝑃 𝐺 𝑛 𝑓 and 𝑖 𝑑 𝑆 𝑃 𝐺 𝑛 𝑋 .
By Theorem 1, 𝑆 𝑃 𝐺 𝑛 𝑋 and 𝑆 𝑃 𝐺 𝑛 𝑌 are homotopically equivalentterm. □ If a set A is a retractterm of the topological space X, then the set SP 𝐺 𝑛 𝐴 is a retractterm of the topological space SP 𝐺 𝑛 𝑋 .
Suppose that a set A is a retractterm of X. Then there exists a continuous mapping 𝑟 : 𝑋 → 𝐴 such that 𝑟 ( 𝑎 ) = 𝑎 for all 𝑎 ∈ 𝐴 . Now we consider the mapping SP 𝐺 𝑛 𝑟 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝐴 . For every [ ( 𝑎 1 , 𝑎 2 , … , 𝑎 𝑛 ) ] 𝐺 ∈ SP 𝐺 𝑛 𝐴 we have SP 𝐺 𝑛 𝑟 ( [ ( 𝑎 1 , 𝑎 2 , … , 𝑎 𝑛 ) ] 𝐺 ) = [ ( 𝑟 ( 𝑎 1 ) , 𝑟 ( 𝑎 2 ) , … , 𝑟 ( 𝑎 𝑛 ) ) ] 𝐺 = [ ( 𝑎 1 , 𝑎 2 , … , 𝑎 𝑛 ) ] 𝐺 .
This means that the mapping SP 𝐺 𝑛 𝑟 : SP 𝐺 𝑛 𝑋 → SP 𝐺 𝑛 𝐴 is a retractionterm. Hence, the set SP 𝐺 𝑛 𝐴 is a retractterm of the space SP 𝐺 𝑛 𝑋 . □ The functor 𝑆 𝑃 𝐺 𝑛 is a covariant homotopy functorterm.
Now we will show that the functor SP 𝐺 𝑛 satisfies the above three conditions.
(i) Let 𝑖 𝑑 𝑋 be identity mapping in the topological space X. Then we have that SP 𝐺 𝑛 𝑖 𝑑 𝑋 [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 = [ ( 𝑖 𝑑 𝑋 ( 𝑥 1 ) , 𝑖 𝑑 𝑋 ( 𝑥 2 ) , … , 𝑖 𝑑 𝑋 ( 𝑥 𝑛 ) ) ] 𝐺 = [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] . This means that the mapping SP 𝐺 𝑛 𝑖 𝑑 𝑋 is the identity mapping in the topological space SP 𝐺 𝑛 𝑋 .
(ii) Let 𝑓 : 𝑋 → 𝑌 , 𝑔 : 𝑌 → 𝑍 be continuous mappings. Then it follows that SP 𝐺 𝑛 ( 𝑔 ∘ 𝑓 ) [ ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) ] 𝐺 = [ ( ( 𝑔 ∘ 𝑓 ) ( 𝑥 1 ) , ( 𝑔 ∘ 𝑓 ) ( 𝑥 2 ) , … , ( 𝑔 ∘ 𝑓 ) ( 𝑥 𝑛 ) ) ] 𝐺 = SP 𝐺 𝑛 𝑔 [ ( 𝑓 ( 𝑥 1 ) , 𝑓 ( 𝑥 2 ) , … , 𝑓 ( 𝑥 𝑛 ) ) ] 𝐺 = SP 𝐺 𝑛 𝑔 ∘ SP 𝐺 𝑛 𝑓 .
(iii) It follows easily from Theorem 1. □ In [14], some propositions about homotopyterm properties of the topological spaces were given. For instance, it was proved that contractibility, connectedness, and pathwise connectedness are homotopyterm property of the spaces.
If a topological space X is contractibleterm, then the space SP 𝐺 𝑛 𝑋 is also contractibleterm.
If a topological space X is connected (viz., pathwise connected), then the space SP 𝐺 𝑛 𝑋 is also connected (viz., pathwise connected).
If f and g are two paths in X with 𝑓 ( 1 ) = 𝑔 ( 0 ) , then by the product of f and g we mean the path 𝑓 ∗ 𝑔 , which is defined by ( 𝑓 ∗ 𝑔 ) ( 𝑡 ) = 𝑓 ( 2 𝑡 ) , if 0 ≤ 𝑡 ≤ 1 / 2 , 𝑔 ( 2 𝑡 − 1 ) , if 1 / 2 ≤ 𝑡 ≤ 1 .
Let 𝑓 ≃ 𝑔 and 𝑔 ≃ ℎ , where F is a homotopyterm from f to g and G is a homotopyterm from g to h. Then, 𝑓 ≃ ℎ . Define a homotopyterm 𝐻 : 𝑋 × 𝐼 → 𝑌 between f and h as follows: 𝐻 ( 𝑥 , 𝑡 ) = 𝐹 ( 𝑥 , 2 𝑡 ) , if 0 ≤ 𝑡 ≤ 1 / 2 , 𝐺 ( 𝑥 , 2 𝑡 − 1 ) , if 1 / 2 ≤ 𝑡 ≤ 1 .
If the mappings 𝑓 𝑖 : 𝐼 → 𝑋 are paths in X from the points 𝑥 0 𝑖 to the points 𝑥 1 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 , respectively, then the mapping SP 𝐺 𝑛 𝑓 𝑛 : 𝐼 → SP 𝐺 𝑛 𝑋 defined by SP 𝐺 𝑛 𝑓 𝑛 ( 𝑡 ) = [ ( 𝑓 1 ( 𝑡 ) , 𝑓 2 ( 𝑡 ) , … , 𝑓 𝑛 ( 𝑡 ) ) ] 𝐺 is also a path from the point [ ( 𝑥 0 1 , 𝑥 0 2 , … , 𝑥 0 𝑛 ) ] 𝐺 to the point [ ( 𝑥 1 1 , 𝑥 1 2 , … , 𝑥 1 𝑛 ) ] 𝐺 in SP 𝐺 𝑛 𝑋 .
Let SP 𝐺 𝑛 𝑓 𝑛 and SP 𝐺 𝑛 𝑔 𝑛 be paths from [ ( 𝑥 0 1 , 𝑥 0 2 , … , 𝑥 0 𝑛 ) ] 𝐺 to [ ( 𝑥 1 1 , 𝑥 1 2 , … , 𝑥 1 𝑛 ) ] 𝐺 and from [ ( 𝑥 2 1 , 𝑥 2 2 , … , 𝑥 2 𝑛 ) ] 𝐺 to [ ( 𝑥 3 1 , 𝑥 3 2 , … , 𝑥 3 𝑛 ) ] 𝐺 , respectively, with SP 𝐺 𝑛 𝑓 𝑛 ( 1 ) = SP 𝐺 𝑛 𝑔 𝑛 ( 0 ) . Then we define the multiplication of the paths in SP 𝐺 𝑛 𝑋 as follows: ( SP 𝐺 𝑛 𝑓 𝑛 ∗ SP 𝐺 𝑛 𝑔 𝑛 ) ( 𝑡 ) = [ ( ( 𝑓 1 ∗ 𝑔 1 ) ( 𝑡 ) , ( 𝑓 2 ∗ 𝑔 2 ) ( 𝑡 ) , … , ( 𝑓 𝑛 ∗ 𝑔 𝑛 ) ( 𝑡 ) ) ] 𝐺 .
This path sharing the points [ ( 𝑥 0 1 , 𝑥 0 2 , … , 𝑥 0 𝑛 ) ] 𝐺 and [ ( 𝑥 3 1 , 𝑥 3 2 , … , 𝑥 3 𝑛 ) ] 𝐺 .
In [18], it is shown that the multiplication of equivalence classes of paths is associative; in other words, ( [ 𝑓 ] [ 𝑔 ] ) [ ℎ ] = [ 𝑓 ] ( [ 𝑔 ] [ ℎ ] ) .
Let f and g be paths from the initial point 𝑥 0 to the final point 𝑥 1 . If there is a homotopyterm F from f to g such that for each 𝑡 ∈ 𝐼 , 𝐹 ( 0 , 𝑡 ) = 𝑥 0 and 𝐹 ( 1 , 𝑡 ) = 𝑥 1 , then f and g are said to be path-homotopic [18]. For a path f, [ 𝑓 ] denotes the equivalence class of all paths path-homotopic to f.
The operation ∗ defined above can be applied to homotopyterm classes as well. Let 𝑓 : 𝐼 → 𝑋 be a path from 𝑥 0 to 𝑥 1 and 𝑔 : 𝐼 → 𝑋 a path from 𝑥 1 to 𝑥 2 . Then, one defines [ 𝑓 ] ∗ [ 𝑔 ] = [ 𝑓 ∗ 𝑔 ] .
Let X be a space and 𝑥 0 𝑖 𝑛 𝑋 . A path in X beginning and ending at 𝑥 0 is called a loop [18] based at 𝑥 0 . Denote by 𝜋 1 ( 𝑋 , 𝑥 0 ) the set of all equivalence classes [ 𝑓 ] of loops in X based at 𝑥 0 . 𝜋 1 ( 𝑋 , 𝑥 0 ) with the operation ∗ is a group, where the identity element of the group is [ 𝑒 𝑥 ] and the inverse element of [ 𝑓 ] is [ 𝑓 ¯ ( 𝑡 ) ] = [ 𝑓 ( 1 − 𝑡 ) ] . We call 𝜋 1 ( 𝑋 , 𝑥 0 ) the fundamental groupterm [18].
Suppose ( 𝐺 , ∗ ) and ( 𝐺 1 , ∗ 1 ) are groups. A homomorphism is a mapping such that 𝑓 ( 𝑥 ∗ 𝑦 ) = 𝑓 ( 𝑥 ) ∗ 1 𝑓 ( 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝐺 . A homomorphism f is called an isomorphism if it is bijective.
The fundamental groupsterm of a space and its quotient space are not always isomorphic.
Let 𝑋 = [ 0 , 1 ] and let 𝑆 1 be the unit circle. Clearly, 𝑆 1 is the quotient space of the space X, where the quotient mapping is defined as 𝑞 ( 𝑥 ) = ( 𝑐 𝑜 𝑠 ( 2 𝜋 𝑥 ) ; 𝑠 𝑖 𝑛 ( 2 𝜋 𝑥 ) ) . We know that 𝑋 = [ 0 , 1 ] has the trivial fundamental groupterm (the group consisting of the identity), and the fundamental groupterm of 𝑆 1 is isomorphic to the group ( 𝑍 , + ) .
The fundamental groupsterm 𝜋 1 ( 𝑋 , 𝑥 0 ) and 𝜋 1 ( 𝑆 𝑃 𝐺 𝑛 𝑋 , [ 𝑥 0 ] 𝐺 ) of the topological spaces X and 𝑆 𝑃 𝐺 𝑛 𝑋 are not always isomorphic for every 𝑥 0 ∈ 𝑋 , where [ 𝑥 0 ] 𝐺 = [ ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) ] 𝐺 = ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) .
Let ℎ : 𝑋 → 𝑌 be a continuous mapping sending the point 𝑥 0 ∈ 𝑋 to the point 𝑦 0 ∈ 𝑌 ; We denote this fact by ℎ : ( 𝑋 , 𝑥 0 ) → ( 𝑌 , 𝑦 0 ) . If f is a loop in X based at 𝑥 0 , then the composition ℎ ∘ 𝑓 : 𝐼 → 𝑌 is a loop in Y based at 𝑦 0 . In this way the correspondence 𝑓 → ℎ ∘ 𝑓 gives rise to a mapping from 𝜋 1 ( 𝑋 , 𝑥 0 ) to 𝜋 1 ( 𝑌 , 𝑦 0 ) . Define ℎ ∗ : 𝜋 1 ( 𝑋 , 𝑥 0 ) → 𝜋 1 ( 𝑌 , 𝑦 0 ) by ℎ ∗ ( [ 𝑓 ] ) = [ ℎ ∘ 𝑓 ] . In [19] it is proved that ℎ ∗ is a homomorphism.
The fundamental groupsterm 𝜋 1 ( 𝑋 , 𝑥 0 ) and 𝜋 1 ( SP 𝐺 𝑛 𝑋 , [ 𝑥 0 ] ) of the topological spaces X and SP 𝐺 𝑛 𝑋 are homomorphic for every 𝑥 0 ∈ 𝑋 , and the homomorphism is defined by ( 𝜋 𝑛 , 𝐺 𝑠 ) ∗ [ 𝑓 ] = [ 𝜋 𝑛 , 𝐺 𝑠 ∘ 𝑓 ] .
We know that 𝜋 𝑛 , 𝐺 𝑠 : 𝑋 𝑛 → 𝑆 𝑃 𝐺 𝑛 𝑋 is a continuous mapping that carries the point ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) of 𝑋 𝑛 to the point [ ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) ] of 𝑆 𝑃 𝐺 𝑛 𝑋 . If f is a loop in X based at 𝑥 0 , then 𝑓 𝑛 is a loop in 𝑋 𝑛 based at ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) and the composition 𝜋 𝑛 , 𝐺 𝑠 ∘ 𝑓 𝑛 : 𝐼 → 𝑆 𝑃 𝐺 𝑛 𝑋 is a loop in 𝑆 𝑃 𝐺 𝑛 𝑋 based at [ ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) ] . The correspondence 𝑓 → 𝜋 𝑛 , 𝐺 𝑠 ∘ 𝑓 𝑛 thus gives rise to a mapping carrying 𝜋 1 ( 𝑋 , 𝑥 0 ) into 𝜋 1 ( 𝑆 𝑃 𝐺 𝑛 𝑋 , [ ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) ] ) . Define 𝜋 𝑛 , 𝐺 𝑠 ∗ : 𝜋 1 ( 𝑋 , 𝑥 0 ) → 𝜋 1 ( 𝑆 𝑃 𝐺 𝑛 𝑋 , [ ( 𝑥 0 , 𝑥 0 , … , 𝑥 0 ) ] ) by the equation 𝜋 𝑛 , 𝐺 𝑠 ∗ ( [ 𝑓 ] ) = [ 𝜋 𝑛 , 𝐺 𝑠 ∘ 𝑓 𝑛 ] . The mapping 𝜋 𝑛 , 𝐺 𝑠 ∗ is a homomorphism (as we said before). □ 5. Conclusions In this article we continue the study of the functor of permutation degreeterm—one of important functors in topology. Our results extend and complement the existing results in this field. We obtained several relations among cardinal invariantsterm in the space SP 2 ( 𝑅 , 𝜏 ( 𝐴 ) ) of the permutation degreeterm of the Hattori spaceterm ( 𝑅 , 𝜏 ( 𝐴 ) ) . These cardinal invariantsterm include the hereditary densityterm, hereditary weak densityterm, spreadterm, extentterm, 𝜋 -weight, and (hereditary) Lindelöf numberterm. Additionally, we proved that if the spaces X and Y are homotopically equivalentterm, then the spaces SP 𝐺 𝑛 𝑋 and SP 𝐺 𝑛 𝑌 are homotopically equivalentterm too. As a consequence, one obtains that the functor SP 𝐺 𝑛 is a covariant homotopy functorterm. It preserves a few topological properties, including retractsterm.
We believe that our results can be applied to similar investigation of other topological properties and other functors.