On r-Compactness in Topological and Bitopological Spaces — TEI-tagged view

Jamal OudetallahpersName, Rehab AlharbipersName, Iqbal M. BatihapersName (2023date). “On r-Compactness in Topological and Bitopological Spaces.” Axiomstitle·j.

Object of study — r-compactness and its pairwise generalisation in topological and bitopological spacesterm, and their relations to other covering and separation properties

Research question — how the r-compactterm·def covering property relates to compactness, near compactness, quasi-H-closedness and S-closedness, and how it generalises to bitopological (pairwise) spaces

Contribution — the definition of pairwise r-compactness and a family of theorems relating r-compactterm·def and pairwise r-compact spacesterm to other compactness variants and to r-separation axioms

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On r-Compactness in Topological and Bitopological Spacesterm Abstract Abstract This paper defines the so-called pairwise r-compactness in topological and bitopological spacesterm. In particular, several inferred properties of the r-compact spacesterm and their connections with other topological and bitopological spacesterm are studied theoretically. As a result, several novel theorems of the r-compact spaceterm are generalized on the pairwise r-compact spaceterm. The results established in this research paper are new in the field of topology. Keywords: pairwise compact spaceterm; pairwise r-compact spaceterm; pairwise extremely disconnectedterm space; bitopological spaceterm MSC: 54D30; 54A05 1. Introduction Compactness possesses a really vital input in topology and so do a few of its lower and more grounded types. One of such types is H-close in which the hypothesis of these spaces was presented in 1929 by P. S. Alexandroff and his colleague [1]. In 1969, M. K. Singal and A. Aathur presented nearly compact spacesterm [2]. In 1976, T. Thompson presented a different kind of compact spaceterm called the s-compact space [3]. Once in a while, a few additional types of compactness have been investigated [4,5]. In recent time, V. V. Tkachuk provided in [6] a self-contained introduction to Cp-theory and general topology, including a unique problem-based introduction to the theory of function spaces and many results and methods related to the Cp-thoery. At the same time, H. H. Kadhem proposed in [7] a new type of compact spacesterm called an r-compact spaceterm, where a topological space ( 𝑋 , 𝜏 ) is said to be an r-compact spaceterm if every regular open coverterm of X has a finite subfamily whose closures cover X. In this article, we introduce several novel theorems of weaker kind related to the compact spacesterm specified by the r-compact spaceterm, a generalization that concerns the pairwise r-compact spaceterm.

The bitopological spaceterm subject might be written as 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) , where 𝛾 1 and 𝛾 2 are two topologies defined on Z [8]. This concept is connected with a former investigation that was carried out on bitopological spacesterm so that each topology can be defined as a set of points that possesses nearby related points and satisfies specific axioms. In [9], Kelly defined each of the pairwise normal, pairwise Hausdorff and pairwise regular spaces with some conventional theorems indicated by Tietze’s extension. An additional work in the bitopological spaceterm field was performed by Kim in [10]. In [11], the concept of ( 𝛼 − 𝛽 ) -level spaces was defined by taking into account the fuzzy bitopological spaceterm concept. As a consequence of that work, a fuzzy ( 𝛼 − 𝛽 ) of a bitopological Hausdorff space was defined and the notion of a fuzzy ( 𝛼 − 𝛽 ) ∗ of a bitopological Hausdorff space was established using the ( 𝛼 − 𝛽 ) ∗ disjoint sets. In [12], with the help of an extended Pythagorean fuzzy topological space, the Pythagorean fuzzy bitopological spaceterm was defined, and several notions were accordingly inferred related to the pairwise Pythagorean fuzzy topological spaces coupled with several relations of their characteristics. In [13], the compact ultrametrics’ range sets were described in regard to its order type. The expandability, near expandability and feeble expandability of a bitopological spaceterm were explained by OudetallahpersName in [14,15,16].

The primary objective of this work is to present and examine a novel kind of pairwise compact spacesterm, which is the so-called pairwise r-compact spaceterm (or simply the p-r-compact space). Accordingly, we derive several novel results related to the r-compact spaceterm that represent generalizations of their corresponding results from the pairwise r-compact spaceterm.

2. Preliminary In this section, we aim to pave the way to our main results by recalling several significant definitions and facts.

If A ⊆ Z and (Z,γ) is a topological space, then it is said that: 1. A is a regular open setterm in Z if A = 𝐴 ¯ 𝑜 . 2. A is a regular closed set in Z iff A = 𝐴 𝑜 ¯ . 3. There exists an open set U such that U ⊆ A ⊆ 𝑈 ¯ iff A is a semiopen setterm in Z.

Suppose ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a bitopological spaceterm so that 𝐴 ⊆ 𝑍 , then it is said that: 1. A is a pairwise regular open setterm if 𝐴 = 𝐼 𝑛 𝑡 𝛾 1 ( 𝐶 𝐿 𝛾 1 ( 𝐴 ) ) and 𝐴 = 𝐼 𝑛 𝑡 𝛾 2 ( 𝐶 𝐿 𝛾 2 ( 𝐴 ) ) . 2. A is a pairwise regular closed set if 𝐴 = 𝐶 𝐿 𝛾 1 ( 𝐼 𝑛 𝑡 𝛾 1 ( 𝐴 ) ) and 𝐴 = 𝐶 𝐿 𝛾 2 ( 𝐼 𝑛 𝑡 𝛾 2 ( 𝐴 ) ) . 3. A is a pairwise semiopen setterm if ∃ an open set u such that 𝑢 𝛾 1 ⊆ 𝐴 ⊆ 𝐶 𝐿 𝛾 1 ( 𝑢 ) and 𝑢 𝛾 2 ⊆ 𝐴 ⊆ 𝐶 𝐿 𝛾 2 ( 𝑢 ) .

Suppose ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a bitopological spaceterm so that 𝐴 ⊆ 𝑍 , then: 1. A is called a pairwise regular closed set if 𝑍 − 𝐴 is a pairwise regular open setterm. 2. A is called a pairwise regular open setterm if 𝑍 − 𝐴 is a pairwise regular closed set.

Let ( 𝑍 , 𝛾 1 , 𝛾 2 ) be a bitopological spaceterm, then every pairwise open set is a pairwise semiopen setterm.

Assume A is a pairwise regular open setterm, so 𝐼 𝑛 𝑡 𝛾 𝑖 ( 𝐴 ) ⊆ 𝐴 ⊆ 𝐶 𝐿 𝛾 𝑖 ( 𝐴 ) , ∀ 𝑖 = 1 , 2 . Consequently, 𝑢 ⊆ 𝐴 ⊆ 𝐶 𝐿 𝛾 𝑖 ( 𝑢 ) , so A is a pairwise semiopen setterm, ∀ 𝑖 = 1 , 2 . □ A space 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is called a pairwise r-compactterm·def if every regular open coverterm of Z possesses a finite subfamily for which its closures cover Z.

([17]). Let ( 𝑍 , 𝛾 ) be a topological space, then Z is said to be extremely disconnected if for every open set A in Z, 𝐴 ¯ is a clopen setterm in Z.

([17]). Let ( 𝑍 , 𝛾 1 , 𝛾 2 ) be a bitopological spaceterm, then Z is said to be pairwise extremely disconnectedterm if for each 𝛾 𝑖 -open set in Z, 𝐴 ¯ is a 𝛾 𝑖 -open set in Z, ∀ 𝑖 = 1 , 2 .

([17]). A topological space ( 𝑍 , 𝛾 ) is called quasi H-closedterm if every open cover of Z possesses a finite subfamily for which its closures cover Z.

A bitopological spaceterm 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is called pairwise quasi H-closedterm if every 𝛾 𝑖 -open cover of Z possesses a finite subfamily for which its closures cover Z, ∀ 𝑖 = 1 , 2 .

A space ( 𝑍 , 𝛾 ) is said to be a nearly compact spaceterm if every open cover possesses a finite subfamily such that the interiors of its closures cover Z.

A bitopological spaceterm 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is called a pairwise nearly compact spaceterm if every 𝛾 𝑖 -open cover of Z possesses a finite subfamily such that the interiors of its closures cover Z, ∀ 𝑖 = 1 , 2 .

A space ( 𝑍 , 𝛾 ) is called S-closedterm if every semiopen cover of Z possesses a finite subfamily for which its closures cover Z.

A bitopological spaceterm 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is called pairwise S-closedterm if every 𝛾 𝑖 -semiopen cover of Z possesses a finite subfamily for which its closures cover Z, ∀ 𝑖 = 1 , 2 .

3. Main Results In this part, different new theorems and properties of the r-compact spacesterm coupled with their relations with other topological and bitopological spacesterm are presented. In other words, we present in what follows the main results of this work.

If 𝑍 = ( 𝑍 , 𝛾 ) is compact, then it is an r-compact spaceterm.

Assume that Z is a compact spaceterm and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a regular open coverterm of Z. Now, as Z is a compact spaceterm, ∃ 𝜔 𝛼 1 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 𝛼 𝑘 . Since 𝜔 𝛼 𝑘 is a regular open setterm in Z for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 , then 𝜔 𝛼 𝑘 = 𝜔 𝛼 𝑛 for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 . Hence, we have: 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 𝛼 𝑘 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝑜 𝛼 𝑘 ⊆ ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 . (1) Now, since 𝜔 𝛼 𝑘 ⊆ 𝑍 for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 , we obtain: ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 ⊆ 𝑍 . (2) Consequently, by using (1) and (2), we obtain that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 , and therefore, Z is r-compact spaceterm. □ Every pairwise compact spaceterm is a pairwise r-compact spaceterm.

Suppose 𝑖 = 1 , 2 , Z is a pairwise compact spaceterm and { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 -regular open cover of Z. Since Z is a pairwise compact spaceterm, ∃ 𝜔 𝛼 1 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 𝛼 𝑘 . Since 𝜔 𝛼 𝑘 is a 𝛾 𝑖 -regular open set in Z for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 , 𝜔 𝛼 𝑘 = 𝜔 𝛼 𝑛 for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 . Hence, we have: 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 𝛼 𝑘 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝑜 𝛼 𝑘 ⊆ ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 . (3) Since 𝜔 𝛼 ⊆ 𝑍 for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 , we have: ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 ⊆ 𝑍 . (4) Now, by using (3) and (4), we obtain that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 , and therefore Z is a pairwise r-compact spaceterm. □ If 𝑍 = ( 𝑍 , 𝛾 ) is nearly compact, then it is an r-compact spaceterm.

Assume that Z is a nearly compact spaceterm and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a regular open coverterm of Z. Then, { 𝜔 𝛼 | 𝛼 ∈ Ω } is an open cover of Z. Now, as Z is a nearly compact spaceterm, ∃ 𝜔 𝛼 1 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝑜 𝛼 𝑘 . Since ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝑜 𝛼 𝑘 ⊆ ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 , we can assert: 𝑍 ⊆ ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 . (5) In the same regard, since 𝜔 𝛼 𝑘 ⊆ 𝑍 for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 , we have: ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 ⊆ 𝑍 . (6) As a result, from (5) and (6), we obtain that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 , and thus Z is an r-compact spaceterm. □ Every pairwise nearly compact spaceterm is a pairwise r-compact spaceterm.

Assume that 𝑖 = 1 , 2 , Z is a pairwise nearly compact spaceterm and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 regular open coverterm of Z. Thus, { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 open cover of Z. Now, as Z is a nearly compact spaceterm, ∃ 𝜔 𝛼 1 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝑜 𝛼 𝑘 . Since ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝑜 𝛼 𝑘 ⊆ ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 , we can confirm: 𝑍 ⊆ ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 . (7) In addition, since 𝜔 𝛼 𝑘 ⊆ 𝑍 for each 𝛼 𝑘 ∈ Ω and for each 𝑘 = 1 , 2 , ⋯ , 𝑛 , we have: ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 ⊆ 𝑍 . (8) Now, based on (7) and (8), we obtain that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 , and hence Z is a pairwise r-compact spaceterm. □ If 𝑍 = ( 𝑍 , 𝛾 ) is a quasi H-closedterm space, then it is an r-compact spaceterm.

Assume that Z is a quasi H-closedterm space and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a regular open coverterm of Z. Therefore, { 𝜔 𝛼 | 𝛼 ∈ Ω } is an open cover of Z. Accordingly, as Z is quasi H-closedterm, ∃ 𝜔 𝛼 1 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 . Thus, Z is an r-compact spaceterm. □ Every pairwise quasi H-closedterm space is a pairwise r-compact spaceterm.

Assume that Z is a pairwise quasi H-closedterm space and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 regular open coverterm of Z. Hence, { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 open cover of Z. As Z is pairwise quasi H-closedterm, ∃ 𝜔 𝛼 1 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝜔 ¯ 𝛼 𝑘 . Thus, Z is a pairwise r-compact spaceterm. □ If 𝑍 = ( 𝑍 , 𝛾 ) is an S-closedterm space, then it is an r-compact spaceterm.

Assume that Z is an S-closedterm space and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a regular open coverterm of Z. Thus, 𝜈 𝛼 is a semiopen setterm, and { 𝜔 𝛼 | 𝛼 ∈ Ω } is a semiopen cover of Z. As Z is an S-closedterm space, ∃ 𝛾 - 𝜔 𝛼 1 , 𝜔 𝛼 2 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑖 = 1 𝑛 𝜈 ¯ 𝛼 𝑖 . Thus, Z is an r-compact spaceterm. □ Every pairwise S-closedterm space is a pairwise r-compact spaceterm.

Assume that 𝑖 = 1 , 2 , Z is a pairwise S-closedterm space and suppose that { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 -regular open cover of Z. Thus, 𝜈 𝛼 is 𝛾 𝑖 -semiopen set, and { 𝜔 𝛼 | 𝛼 ∈ Ω } is a 𝛾 𝑖 semiopen cover of Z. Now, since Z is a pairwise S-closedterm, there exist 𝛾 𝑖 - 𝜔 𝛼 1 , 𝜔 𝛼 2 , ⋯ , 𝜔 𝛼 𝑛 such that 𝑍 = ⋃ 𝑖 = 1 𝑛 𝜈 ¯ 𝛼 𝑖 , and thus, Z is a pairwise r-compact spaceterm. □ Next, in light of the extremely disconnected spaceterm definition, we intend to state and prove the following results.

If 𝑍 = ( 𝑍 , 𝛾 ) is an extremely disconnected spaceterm, then the statements below are equivalent: 1. Z is r-compactterm·def. 2. Z is nearly compact. 3. Z is quasi-H-closed.

1 → 2 : Suppose 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a open cover of Z. As Z is a pairwise r-compact spaceterm, ∃ 𝐵 ∼ = { 𝑢 𝛼 𝑘 : 𝑘 = 1 , 2 , ⋯ , 𝑛 } for which 𝑍 = ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ and 𝑢 𝛼 𝑘 ∈ 𝑈 , ∀ 𝑖 = 1 , ⋯ , 𝑛 . Moreover, since Z is an extremely disconnected spaceterm, 𝑢 𝛼 𝑘 ¯ is an open set, ∀ 𝑘 = 1 , ⋯ , 𝑛 . Therefore, 𝑢 𝛼 𝑘 ¯ 𝑜 = 𝑢 𝛼 𝑘 ¯ , and so 𝑍 = ⋃ 𝑘 = 1 𝑛 𝑢 ¯ 𝑜 𝛼 𝑘 . Thus, 𝑢 ¯ 𝑜 𝛼 𝑘 𝑘 = 1 𝑛 forms a subfamily of interior sets covered by Z, and therefore Z is nearly compact. 2 → 3 : Assume that 𝑍 = ( 𝑍 , 𝛾 ) is nearly compact. Suppose 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a open cover of Z. Now, as Z is nearly compact, ∃ a finite subfamily 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 and 𝑍 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ 𝑜 . Nevertheless, ∀ 𝑘 = 1 , ⋯ , 𝑛 and ∀ 𝛼 ∈ Λ , we have 𝑢 𝛼 𝑘 ¯ 𝑜 ∈ 𝑢 𝛼 𝑘 ¯ . Thus, 𝑍 ∈ 𝑢 𝛼 𝑘 ¯ 𝑜 ∈ 𝑢 𝛼 𝑘 ¯ , and so there exists a subfamily 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 of 𝑈 ∼ whose closures cover Z. Therefore, Z is a quasi-H-closed space. 3 → 1 : Suppose Z is quasi-H-closed. Assume that 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a cover of Z, where 𝑢 𝛼 is a regular open setterm. Now, since Z is a quasi-H-closed space, 𝑈 ∼ has a finite subfamily 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 such that 𝑍 ∈ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ . Hence, Z is an r-compact spaceterm.

If 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a pairwise extremely disconnectedterm space, then the statements below are equivalent: 1. Z is pairwise r-compactterm·def. 2. Z is pairwise nearly compact. 3. Z is pairwise quasi-H-closed.

1 → 2 : Assume that 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a 𝛾 𝑖 -open cover of Z, ∀ 𝑖 = 1 , 2 . Since Z is a pairwise r-compact spaceterm, ∃ 𝐵 ∼ = { 𝑢 𝛼 𝑘 : 𝑘 = 1 , 2 , ⋯ , 𝑛 } such that 𝑍 = ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ and 𝑢 𝛼 𝑘 ∈ 𝑈 , ∀ 𝑘 = 1 , ⋯ , 𝑛 . Furthermore, since Z is a pairwise extremely disconnectedterm space, 𝑢 𝛼 𝑘 ¯ is an open set, ∀ 𝑘 = 1 , ⋯ , 𝑛 . Consequently, we obtain 𝑢 𝛼 𝑘 ¯ 𝑜 = 𝑢 𝛼 𝑘 ¯ , and so 𝑍 = ⋃ 𝑘 = 1 𝑛 𝑢 ¯ 𝑜 𝛼 𝑘 . This implies 𝑢 ¯ 𝑜 𝛼 𝑘 𝑘 = 1 𝑛 form a subfamily of interior sets that cover Z, and therefore Z is pairwise nearly compact. 2 → 3 : Suppose that 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is pairwise nearly compact. Assume that 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a 𝛾 𝑖 -open cover of Z, ∀ 𝑖 = 1 , 2 . Since Z is pairwise nearly compact, there exists a 𝛾 𝑖 -finite subfamily 𝑢 𝛼 𝑘 : ∀ 𝑘 = 1 , ⋯ , 𝑛 and 𝑍 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ 𝑜 . Nevertheless, ∀ 𝑘 = 1 , ⋯ , 𝑛 and ∀ 𝛼 ∈ Λ , we have 𝑢 𝛼 𝑘 ¯ 𝑜 ∈ 𝑢 𝛼 𝑘 ¯ . Thus, 𝑍 ∈ 𝑢 𝛼 𝑘 ¯ 𝑜 ∈ 𝑢 𝛼 𝑘 ¯ , and so ∃ a subfamily 𝑢 𝛼 𝑘 : ∀ 𝑘 = 1 , ⋯ , 𝑛 of 𝑈 ∼ whose closures cover Z. Hence, Z is a pairwise quasi-H-closed space. 3 → 1 : Suppose Z is pairwise quasi-H-closed. Let 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ }, where 𝑢 𝛼 is a regular open setterm, be a 𝛾 𝑖 -cover of Z, ∀ 𝑖 = 1 , 2 . Now, since Z is a pairwise quasi-H-closed space, 𝑈 ∼ has a 𝛾 𝑖 finite subfamily 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 such that 𝑍 ∈ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ . So, Z is a pairwise r-compact spaceterm.

Let 𝑍 = ( 𝑍 , 𝛾 ) be an r-compactterm·def and extremely disconnected spaceterm, then every closed subspaceterm of Z is r-compactterm·def.

Suppose 𝑍 = ( 𝑍 , 𝛾 ) is an r-compactterm·def and extremely disconnected spaceterm and let ( 𝐴 , 𝛾 𝐴 ) be a subspace of Z. First, to show A is r-compactterm·def, we assume that 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a regular open coverterm of A. This implies 𝑢 𝛼 ¯ 𝑜 = 𝑢 𝛼 , ∀ 𝛼 ∈ Λ and 𝐴 ⊆ ⋃ 𝛼 ∈ Λ 𝑢 𝛼 . Now, 𝑍 = 𝐴 ∪ ( 𝑍 − 𝐴 ) , and so 𝑍 ⊆ ⋃ 𝛼 ∈ Λ 𝑢 𝛼 ∪ ( 𝑍 − 𝐴 ) . In this regard, 𝑍 − 𝐴 is open in Z and Z is extremely disconnected. Then, 𝑍 − 𝐴 is a clopen setterm in Z, and therefore, 𝑍 − 𝐴 ¯ 𝑜 = 𝑍 − 𝐴 . Consequently, 𝑍 − 𝐴 is a regular open setterm, hence 𝑈 ∼ ∗ = 𝑢 𝛼 : 𝛼 ∈ Λ , 𝑍 − 𝐴 forms a regular open coverterm of Z. In addition, since Z is an r-compact spaceterm, then 𝑈 ∼ ∗ has a finite subfamily of the form 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 , 𝑍 − 𝐴 such that 𝑍 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ ∪ 𝑍 − 𝐴 ¯ . Now, 𝑍 − 𝐴 ¯ covers 𝑍 − 𝐴 , and so 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ . Thus, 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 is a finite subfamily of 𝑈 ∼ whose closures cover A. Hence, A is r-compactterm·def. □ Let 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) be a pairwise r-compactterm·def and a pairwise extremely disconnectedterm space, then every closed subspaceterm of Z is pairwise r-compactterm·def.

Suppose 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a pairwise r-compactterm·def and pairwise extremely disconnectedterm space and let ( 𝐴 , 𝛾 1 𝐴 , 𝛾 2 𝐴 ) be a subspace of Z. Herein, to show A is pairwise r-compactterm·def, we assume that 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } is a 𝛾 𝑖 -regular open cover of A, ∀ 𝑖 = 1 , 2 . This gives 𝑢 𝛼 ¯ 𝑜 = 𝑢 𝛼 , ∀ 𝛼 ∈ Λ and 𝐴 ⊆ ⋃ 𝛼 ∈ Λ 𝑢 𝛼 . Now, 𝑍 = 𝐴 ∪ ( 𝑍 − 𝐴 ) , and so 𝑍 ⊆ ⋃ 𝛼 ∈ Λ 𝑢 𝛼 ∪ ( 𝑍 − 𝐴 ) . In this respect, 𝑍 − 𝐴 is 𝛾 𝑖 -open in Z and Z is pairwise extremely disconnectedterm. Therefore, 𝑍 − 𝐴 is a 𝛾 𝑖 -clopen set in Z, and then 𝑍 − 𝐴 ¯ 𝑜 = 𝑍 − 𝐴 . Thus, 𝑍 − 𝐴 is a 𝛾 𝑖 -regular open set, and 𝑈 ∼ ∗ = 𝑢 𝛼 : 𝛼 ∈ Λ , 𝑍 − 𝐴 form a regular 𝛾 𝑖 -open cover of Z. Moreover, as Z is a 𝛾 𝑖 -r-compact space, 𝑈 ∼ ∗ has a 𝛾 𝑖 -finite subfamily of the type 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 such that 𝑍 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ ∪ 𝑍 − 𝐴 ¯ . Now, 𝑍 − 𝐴 ¯ covers 𝑍 − 𝐴 , so 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ¯ . Thus, 𝑢 𝛼 𝑘 : 𝑘 = 1 , ⋯ , 𝑛 is a 𝛾 𝑖 -finite subfamily of 𝑈 ∼ whose closures cover A. Hence, A is pairwise r-compactterm·def. □ Assume that 𝑍 = ( 𝑍 , 𝛾 ) is a topological space, then Z is called: 1. An r- 𝑇 𝑜 -space if ∀ 𝑥 ≠ 𝑦 in Z, ∃ a regular open setterm that includes one of them but not both. 2. An r- 𝑇 1 -space if ∀ 𝑥 ≠ 𝑦 in Z, ∃ two regular open setsterm 𝑢 𝑥 and 𝑣 𝑦 so that 𝑥 ∈ 𝑢 𝑥 , 𝑦 ∉ 𝑢 𝑥 and 𝑥 ∉ 𝑣 𝑦 , 𝑦 ∈ 𝑣 𝑦 . 3. An r- 𝑇 2 -space if ∀ 𝑥 ≠ 𝑦 in Z, ∃ two disjoint regular open setsterm 𝑢 𝑥 and 𝑣 𝑦 so that 𝑥 ∈ 𝑢 𝑥 and 𝑣 ∈ 𝑣 𝑦 . 4. An r-regular spaceterm if ∀ 𝑥 ∉ 𝐴 and A a closed set in Z, ∃ two disjoint regular open setsterm 𝑢 𝑥 and 𝑣 𝑦 so that 𝑥 ∈ 𝑢 𝑥 and 𝐴 ⊆ 𝑣 𝑦 . 5. An r- 𝑇 3 -space if Z is an r- 𝑇 1 -space and an r-regular spaceterm. 6. An r-normal spaceterm if for every two disjoint closed sets A and B in Z, ∃ two disjoint regular open setsterm 𝑢 𝐴 and 𝑣 𝐵 so that 𝐴 ⊆ 𝑢 𝐴 and 𝐵 ⊆ 𝑣 𝐵 . 7. An r- 𝑇 4 -space if Z is an r-normal spaceterm and an r- 𝑇 1 space.

Assume that 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a bitopological spaceterm, then Z is called: 1. A pairwise r- 𝑇 𝑜 -space if ∀ 𝑥 ≠ 𝑦 in Z, ∃ a 𝛾 𝑖 -regular open set that includes one of them but not both. 2. A pairwise r- 𝑇 1 -space if ∀ 𝑥 ≠ 𝑦 in Z, ∃ two 𝛾 𝑖 -regular open sets 𝑢 𝑥 and 𝑣 𝑦 so that 𝑥 ∈ 𝑢 𝑥 , 𝑦 ∉ 𝑢 𝑥 and 𝑥 ∉ 𝑣 𝑦 , 𝑦 ∈ 𝑣 𝑦 . 3. A pairwise r- 𝑇 2 -space if ∀ 𝑥 ≠ 𝑦 in Z, ∃ two disjoint 𝛾 𝑖 -regular open sets 𝑢 𝑥 and 𝑣 𝑦 so that 𝑥 ∈ 𝑢 𝑥 and 𝑣 ∈ 𝑣 𝑦 . 4. A pairwise r-regular spaceterm if ∀ 𝑥 ∉ 𝐴 and A a 𝛾 𝑖 -closed set in Z, ∃ two disjoint 𝛾 𝑖 -regular open sets 𝑢 𝑥 and 𝑣 𝑦 so that 𝑥 ∈ 𝑢 𝑥 and 𝐴 ⊆ 𝑣 𝑦 . 5. A pairwise r- 𝑇 3 -space if Z is a pairwise r- 𝑇 1 space and a pairwise r-regular spaceterm. 6. A pairwise r-normal spaceterm if for every two disjoint 𝛾 𝑖 -closed sets A and B in Z, ∃ two disjoint 𝛾 𝑖 -regular open sets 𝑢 𝐴 and 𝑣 𝐵 so that 𝐴 ⊆ 𝑢 𝐴 and 𝐵 ⊆ 𝑣 𝐵 . 7. A pairwise r- 𝑇 4 -space if Z is a pairwise r-normal spaceterm and a pairwise r- 𝑇 1 space.

If A is an r-compactterm·def subset in a r- 𝑇 2 -space, then ∀ 𝑥 ∉ 𝐴 , ∃ two disjoint r-open sets 𝑢 𝑥 and 𝑣 𝐴 so that 𝑥 ∈ 𝑢 𝑥 and 𝐴 ⊆ 𝑣 𝐴 .

For all 𝑎 ∈ 𝐴 , we have 𝑎 ≠ 𝑥 . Now, since 𝑥 ∉ 𝐴 and Z is an r- 𝑇 2 -space, ∃ two r-open sets 𝑢 𝑥 ( 𝑎 ) and 𝑣 ( 𝑎 ) for which 𝑥 ∈ 𝑢 𝑥 ( 𝑎 ) , 𝑎 ∈ 𝑣 ( 𝑎 ) and 𝑢 𝑥 ( 𝑎 ) ∩ 𝑣 ( 𝑎 ) = 𝜙 . Now, 𝑉 ∼ = { 𝑣 ( 𝑎 ) : 𝑎 ∈ 𝐴 } forms an r-open cover of A. However, A is r-compactterm·def, and thus 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 ( 𝑎 𝑘 ) . So, for all v-open sets 𝑣 ( 𝑎 𝑘 ) , 𝑘 = 1 , ⋯ , 𝑛 , ∃ a corresponding open set 𝑢 𝑎 𝑘 ( 𝑥 ) , such that 𝑥 ∈ 𝑢 𝑎 𝑘 ( 𝑥 ) and 𝑢 𝑎 𝑘 ( 𝑥 ) ∩ 𝑣 ( 𝑎 𝑘 ) = 𝜙 as Z is an r- 𝑇 2 -space. Now, let 𝑈 ∼ = ⋂ 𝑘 = 1 𝑛 𝑢 𝑎 𝑘 ( 𝑥 ) . Then, u is an r-open set in Z whereby 𝑢 ∪ 𝑣 ( 𝑎 𝑘 ) = 𝜙 , ∀ 𝑘 = 1 , ⋯ , 𝑛 . Hence, 𝑢 ⊆ 𝑣 ( 𝑎 𝑘 ) implies that 𝑢 ∩ 𝑣 ( 𝑎 𝑘 ) ⊆ 𝑢 𝑎 𝑘 ( 𝑥 ) ∩ 𝑣 ( 𝑎 𝑘 ) = 𝜙 , and therefore, ( 𝑢 ∩ 𝑣 ( 𝑎 1 ) ) ∪ ( 𝑢 ∩ 𝑣 ( 𝑎 2 ) ) ∪ ⋯ ∪ ( 𝑢 ∩ 𝑣 ( 𝑎 𝑛 ) ) = 𝜙 . Thus, 𝑢 ∩ ⋃ 𝑘 = 1 𝑛 𝑣 ( 𝑎 𝑘 ) = 𝜙 . Assume that 𝑟 = ⋃ 𝑘 = 1 𝑛 𝑣 ( 𝑎 𝑘 ) . Thus, v is an r-open set in Z with 𝐴 ∈ 𝑟 , and so the result holds. □ If A is a 𝛾 𝑖 -r-compact subset in a pairwise r- 𝑇 2 -space, then ∀ 𝑥 ∉ 𝐴 , ∃ two disjoint 𝛾 𝑖 -r-open sets 𝑢 𝑥 and 𝑣 𝐴 such that 𝑥 ∈ 𝑢 𝑥 and 𝐴 ⊆ 𝑣 𝐴 .

Let 𝑖 = 1 , 2 . For all 𝑎 ∈ 𝐴 , we get 𝑎 ≠ 𝑥 . Since 𝑥 ∉ 𝐴 and Z is a pairwise r- 𝑇 2 -space, ∃ two 𝛾 𝑖 -r-open sets 𝑢 𝑥 ( 𝑎 ) and 𝑣 ( 𝑎 ) for which 𝑥 ∈ 𝑢 𝑥 ( 𝑎 ) , 𝑎 ∈ 𝑣 ( 𝑎 ) and 𝑢 𝑥 ( 𝑎 ) ∩ 𝑣 ( 𝑎 ) = 𝜙 . Now, 𝑉 ∼ = { 𝑣 ( 𝑎 ) : 𝑎 ∈ 𝐴 } forms a 𝛾 𝑖 -r-open cover of A. However, A is 𝛾 𝑖 -r-compact, and thus 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 ( 𝑎 𝑘 ) . Thus, ∀ 𝛾 𝑖 -r-open sets 𝑣 ( 𝑎 𝑘 ) , 𝑘 = 1 , ⋯ , 𝑛 , ∃ a corresponding 𝛾 𝑖 -r-open set 𝑢 𝑎 𝑘 ( 𝑥 ) for which 𝑥 ∈ 𝑢 𝑎 𝑘 ( 𝑥 ) and 𝑢 𝑎 𝑘 ( 𝑥 ) ∩ 𝑣 ( 𝑎 𝑘 ) = 𝜙 because Z is a pairwise r- 𝑇 2 -space. Now, let 𝑈 ∼ = ⋂ 𝑘 = 1 𝑛 𝑢 𝑎 𝑘 ( 𝑥 ) . Then, u is a 𝛾 𝑖 -r-open set in Z with 𝑢 ∪ 𝑣 ( 𝑎 𝑘 ) = 𝜙 , ∀ 𝑘 = 1 , ⋯ , 𝑛 . Hence, 𝑢 ⊆ 𝑣 ( 𝑎 𝑘 ) implies that 𝑢 ∩ 𝑣 ( 𝑎 𝑘 ) ⊆ 𝑢 𝑎 𝑘 ( 𝑥 ) ∩ 𝑣 ( 𝑎 𝑘 ) = 𝜙 , and therefore, ( 𝑢 ∩ 𝑣 ( 𝑎 1 ) ) ∪ ( 𝑢 ∩ 𝑣 ( 𝑎 2 ) ) ∪ ⋯ ∪ ( 𝑢 ∩ 𝑣 ( 𝑎 𝑛 ) ) = 𝜙 . Thus, 𝑢 ∩ ⋃ 𝑘 = 1 𝑛 𝑣 ( 𝑎 𝑘 ) = 𝜙 . Now, if one lets 𝑟 = ⋃ 𝑘 = 1 𝑛 𝑣 ( 𝑎 𝑘 ) , then r is a 𝛾 𝑖 -r-open set in Z and 𝐴 ∈ 𝑟 , and so the result holds. □ If A and B are two disjoint r-compactterm·def subsets of an r- 𝑇 2 -space 𝑍 = ( 𝑍 , 𝛾 ) , then ∃ two disjoint r-open sets 𝑢 𝐴 and 𝑣 𝐵 so that 𝐴 ⊆ 𝑢 𝐴 and 𝐵 ⊆ 𝑣 𝐵 .

Note that 𝑥 ∈ 𝐴 and 𝑥 ∉ 𝐵 because 𝐴 ∩ 𝐵 = 𝜙 . Therefore, by Theorem 15, ∃ two r-open sets 𝑢 𝑥 and 𝑣 𝑦 in Z for which 𝑥 ∈ 𝑢 𝛼 , 𝐵 ⊆ 𝑣 𝛼 and 𝑢 𝛼 ∩ 𝑣 𝛼 = 𝜙 . Now, 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } forms an r-open cover of A. However, A is an r-compact spaceterm, and so 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 . Now, if one lets 𝑢 = ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 , then u is r-open in Z, and hence we have: 𝐴 ⊆ 𝑢 . (9) Now, ∀ 𝑘 = 1 , ⋯ , 𝑛 , we have an r-open set 𝑢 𝛼 𝑘 that corresponds to an r-open set 𝑣 𝛼 𝑘 so that 𝐵 ⊆ 𝑣 𝛼 𝑘 . Thus, 𝐵 ⊆ ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 . If one lets 𝑣 = ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 , then v is an r-open set such that 𝐵 ⊆ 𝑣 . (10) Now, ∀ 𝑘 = 1 , ⋯ , 𝑛 , we have 𝑢 𝛼 𝑘 ∩ 𝑣 𝛼 𝑘 = 𝜙 . Thus, ( ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ) ∩ 𝑣 𝛼 𝑘 = 𝜙 . Consequently, since ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 ⊆ 𝑣 𝛼 𝑘 , 𝑢 ∩ ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 ⊆ ( 𝑢 ∩ 𝑣 𝛼 𝑘 ) = 𝜙 . Therefore, 𝑢 ∩ 𝑣 ⊆ 𝜙 , hence we have: 𝑢 ∩ 𝑣 = 𝜙 . (11) Consequently, by (9)–(11), the result holds. □ If A and B are two disjoint 𝛾 𝑖 -r-compact subsets of a pairwise r- 𝑇 2 -space 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) , then ∃ two disjoint 𝛾 𝑖 -r-open sets 𝑢 𝐴 and 𝑣 𝐵 so that 𝐴 ⊆ 𝑢 𝐴 and 𝐵 ⊆ 𝑣 𝐵 .

Let 𝑖 = 1 , 2 . For all 𝑥 ∈ 𝐴 , 𝑥 ∉ 𝐵 because 𝐴 ∩ 𝐵 = 𝜙 . Therefore, by Theorem 15, ∃ two r-open sets 𝑢 𝑥 and 𝑣 𝑦 in Z for which 𝑥 ∈ 𝑢 𝛼 , 𝐵 ⊆ 𝑣 𝛼 and 𝑢 𝛼 ∩ 𝑣 𝛼 = 𝜙 . Now, 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } forms an pairwise r-open cover of A. However, A is an r-compact spaceterm, and so 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 . If one lets 𝑢 = ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 , then u is 𝛾 𝑖 -r-open in Z, and hence we have: 𝐴 ⊆ 𝑢 . (12) Now, ∀ 𝑘 = 1 , ⋯ , 𝑛 , we have a 𝛾 𝑖 -r-open set 𝑢 𝛼 𝑘 that corresponds to a 𝛾 𝑖 -r-open set 𝑣 𝛼 𝑘 for which 𝐵 ⊆ 𝑣 𝛼 𝑘 . Thus, 𝐵 ⊆ ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 . Now, if one lets 𝑣 = ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 , then v is a 𝛾 𝑖 -r-open set, and hence we have: 𝐵 ⊆ 𝑣 . (13) Now, ∀ 𝑘 = 1 , ⋯ , 𝑛 , we have 𝑢 𝛼 𝑘 ∩ 𝑣 𝛼 𝑘 = 𝜙 . So, ( ⋃ 𝑘 = 1 𝑛 𝑢 𝛼 𝑘 ) ∩ 𝑣 𝛼 𝑘 = 𝜙 . Immediately, since ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 ⊆ 𝑣 𝛼 𝑘 , 𝑢 ∩ ⋂ 𝑘 = 1 𝑛 𝑣 𝛼 𝑘 ⊆ ( 𝑢 ∩ 𝑣 𝛼 𝑘 ) = 𝜙 . Therefore, 𝑢 ∩ 𝑣 ⊆ 𝜙 , and hence we have: 𝑢 ∩ 𝑣 = 𝜙 . (14) Consequently, by (12), (13) and (14), the result holds. □ If 𝑍 = ( 𝑍 , 𝛾 ) is an r-compactterm·def, r- 𝑇 2 - and r-extremely disconnected space, then Z is an r- 𝑇 4 -space.

Assume that 𝑍 = ( 𝑍 , 𝛾 ) is an r-compactterm·def and r- 𝑇 2 -space, then it is clearly r- 𝑇 1 -space. Now, assume that A and B are two r-closed subsets in Z, in which 𝐴 ∩ 𝐵 = 𝜙 . As Z is an r-compact spaceterm, then, by Theorem 13, A and B are r-compactterm·def subsets in the r- 𝑇 2 -space Z. Thus, by Theorem 17, ∃ two r-open sets 𝑢 𝐴 and 𝑣 𝐵 for which 𝐴 ⊆ 𝑢 𝐴 , 𝐵 ⊆ 𝑣 𝐵 and 𝑢 𝐴 ∩ 𝑣 𝐵 = 𝜙 . Thus, Z is an r- 𝑇 4 -space. □ If 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a pairwise r-compactterm·def, pairwise r- 𝑇 2 - and pairwise r-extremely disconnected space, then Z is a pairwise r- 𝑇 4 -space.

Suppose 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a pairwise r-compactterm·def and pairwise r- 𝑇 2 -space, then it is clearly pairwise r- 𝑇 1 -space. Now, assume that A and B are two 𝛾 𝑖 -r-closed subsets in Z in which 𝐴 ∩ 𝐵 = 𝜙 , ∀ 𝑖 = 1 , 2 . Since Z is a pairwise r-compact spaceterm, by Theorem 13, A and B are two 𝛾 𝑖 -r-compact subsets in the pairwise r- 𝑇 2 -space Z. Thus, by Theorem 17, ∃ two 𝛾 𝑖 -r-open sets 𝑢 𝐴 and 𝑣 𝐵 for which 𝐴 ⊆ 𝑢 𝐴 , 𝐵 ⊆ 𝑣 𝐵 and 𝑢 𝐴 ∩ 𝑣 𝐵 = 𝜙 . Hence, Z is a pairwise r- 𝑇 4 -space. □ Let 𝑍 = ( 𝑍 , 𝛾 ) be an r- 𝑇 2 - and r-extremely disconnected space, then every subset in Z is a closed set.

Assume that A is an r-compactterm·def subset of Z. Now, if one lets 𝑥 ∉ 𝐴 , then by Theorem 14, ∃ two r-open sets 𝑢 𝑥 and 𝑣 𝐴 such that 𝑥 ∈ 𝑢 𝑥 , 𝐴 ⊆ 𝑣 𝐴 and 𝑢 𝑥 ∩ 𝑣 𝐴 = 𝜙 . Thus, 𝑢 ⊆ ( 𝑣 𝐴 ) ∁ . Moreover, since 𝐴 ⊆ 𝑣 implies that 𝑣 ∁ ⊆ 𝐴 ∁ , then 𝑥 ∈ 𝑢 𝑥 ⊆ 𝑣 ∁ ⊂ 𝐴 ∁ , and so u is an r-open set. Hence, 𝐴 ∁ is an r-open set, and so A is an r-closed set. □ Let 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) be a pairwise r- 𝑇 2 - and a pairwise r-extremely disconnected space, then every subset in Z is a 𝛾 𝑖 -closed set.

Assume that A is a 𝛾 𝑖 -r-compact subset of Z. Let 𝑥 ∉ 𝐴 . Then, by Theorem 15, ∃ two 𝛾 𝑖 -r-open sets 𝑢 𝑥 and 𝑣 𝐴 for which 𝑥 ∈ 𝑢 𝑥 , 𝐴 ⊆ 𝑣 𝐴 and 𝑢 𝑥 ∩ 𝑣 𝐴 = 𝜙 . Thus, 𝑢 ⊆ ( 𝑣 𝐴 ) ∁ and since 𝐴 ⊆ 𝑣 implies that 𝑣 ∁ ⊆ 𝐴 ∁ , 𝑥 ∈ 𝑢 𝑥 ⊆ 𝑣 ∁ ⊂ 𝐴 ∁ . Now, since u is a 𝛾 𝑖 -r-open set, 𝐴 ∁ is a 𝛾 𝑖 -r-open set as well. Therefore, A is a 𝛾 𝑖 -r-closed set, ∀ 𝑖 = 1 , 2 . □ If 𝑍 = ( 𝑍 , 𝛾 ) is an r-compactterm·def, r- 𝑇 2 - and r-extremely disconnected space, then every subset of Z is r-compactterm·def if and only if it is an r-closed set.

⇒ ) Assume that A is an r-compactterm·def subset of Z, then by Theorem 20, A is an r-closed set.

⇐ ) Assume that A is an r-closed set in an r-compactterm·def r- 𝑇 2 -extremely disconnected space, then, by Theorem 12, A is r-compactterm·def. □ If 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a pairwise r-compactterm·def, pairwise r- 𝑇 2 - and pairwise r-extremely disconnected space, then every 𝛾 𝑖 -subset of Z is pairwise r-compactterm·def if and only if it is a 𝛾 𝑖 -r-closed set, ∀ 𝑖 = 1 , 2 .

⇒ ) Assume that A is a pairwise r-compactterm·def subset of Z, then by Theorem 21, A is a 𝛾 𝑖 -r-closed set, ∀ 𝑖 = 1 , 2 .

⇐ ) Assume that A is 𝛾 𝑖 -r-closed in a pairwise r-compactterm·def, pairwise r- 𝑇 2 -extremely disconnected space, then, by Theorem 12, A is 𝛾 𝑖 -r-compact, ∀ 𝑖 = 1 , 2 . □ Let 𝑍 = ( 𝑍 , 𝛾 ) be an r- 𝑇 3 -extremely disconnected space. If A is a subset in Z such that 𝐴 ⊆ 𝑢 𝐴 , for some r-open set 𝑢 𝐴 in Z, then ∃ an r-open set 𝑣 𝐴 in Z in which: 𝐴 ⊆ 𝑣 𝐴 ⊆ 𝑣 𝐴 ¯ ⊆ 𝑢 𝐴 .

For all 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝑢 𝐴 , we have Z is r-regular spaceterm. This implies that ∃ an r-open set 𝑣 𝑥 for which: 𝑥 ∈ 𝑣 𝑥 ⊆ 𝑣 𝑥 ¯ ⊆ 𝑢 𝐴 . (15) Now, 𝑉 ∼ = { 𝑣 ( 𝑥 ) : 𝑥 ∈ Λ } forms an r-open cover of A, and because A is r-compactterm·def, 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 . Thus, from (15), we have: 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 ¯ ⊆ 𝑢 𝐴 . Now, if one lets 𝑣 𝐴 = ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 , then we get: 𝐴 ⊆ 𝑣 𝐴 ⊆ 𝑣 𝐴 ¯ ⊆ 𝑢 𝐴 . □ Let 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) be a pairwise r- 𝑇 3 -extremely disconnected space. If A is a subset in Z such that 𝐴 ⊆ 𝑢 𝐴 , for some 𝛾 𝑖 -r-open set 𝑢 𝐴 in Z, then ∃ a 𝛾 𝑖 -r-open set 𝑣 𝐴 in Z in which: 𝐴 ⊆ 𝑣 𝐴 ⊆ 𝑣 𝐴 ¯ ⊆ 𝑢 𝐴 , ∀ 𝑖 = 1 , 2 .

For all 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝑢 𝐴 , we have Z is a pairwise r-regular spaceterm. This consequently implies that ∃ a 𝛾 𝑖 -r-open set 𝑣 𝑥 in which: 𝑥 ∈ 𝑣 𝑥 ⊆ 𝑣 𝑥 ¯ ⊆ 𝑢 𝐴 . (16) Now, 𝑉 ∼ = { 𝑣 ( 𝑥 ) : 𝑥 ∈ Λ } forms a 𝛾 𝑖 -r-open cover of A, and because A is 𝛾 𝑖 -r-compact, 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 , ∀ 𝑖 = 1 , 2 . Thus, based on (16), we have: 𝐴 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 ⊆ ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 ¯ ⊆ 𝑢 𝐴 . Now, if one lets 𝑣 𝐴 = ⋃ 𝑘 = 1 𝑛 𝑣 𝑥 𝑘 , then we get: 𝐴 ⊆ 𝑣 𝐴 ⊆ 𝑣 𝐴 ¯ ⊆ 𝑢 𝐴 . □ Let 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) be a bitopological spaceterm and 𝐴 ∼ be a family of 𝛾 𝑖 subsets of Z. Then, it is said that 𝐴 ∼ has a finite intersection property (F.I.P.) if the intersection of a finite number of members of 𝐴 ∼ is not empty, ∀ 𝑖 = 1 , 2 .

A topological space 𝑍 = ( 𝑍 , 𝛾 ) is an r-extremely disconnected compact spaceterm if and only if every family of closed subsets of Z with the F.I.P has a nonempty intersection.

⇒ ) Suppose Z is an r-extremely disconnected compact spaceterm. Suppose that ∃ a family 𝐹 ∼ = { 𝐹 𝛼 : 𝛼 ∈ Λ } of r-closed subsets of Z with the F.I.P. and ⋂ 𝛼 ∈ Λ 𝐹 𝛼 = 𝜙 . Thus, we have: ⋃ 𝛼 ∈ Λ 𝐹 𝛼 ∁ = ( ⋂ 𝛼 ∈ Λ 𝐹 𝛼 ) ∁ = 𝑍 . However, 𝐹 𝛼 is r-closed in Z, ∀ 𝛼 ∈ Λ . Therefore, 𝐹 𝛼 ∁ is r-open in Z, ∀ 𝛼 ∈ Λ . Thus, 𝑉 ∼ = 𝐹 𝛼 ∁ such that 𝛼 ∈ Λ , for an r-open cover of Z. Therefore, by the assumption 𝑍 = ⋃ 𝑘 = 1 𝑛 𝐹 𝛼 𝑘 ∁ = ( ⋂ 𝑘 = 1 𝑛 𝐹 𝛼 𝑘 ) ∁ , we can have 𝜙 = ⋂ 𝑘 = 1 𝑛 𝐹 𝛼 𝑘 , which contradicts with 𝐹 ∼ that has the F.I.P., and so the result holds.

⇐ ) Suppose every family of r-closed subsets of Z with the F.I.P. has a nonempty intersection. Assume that Z is not an r-compact spaceterm. Then, ∃ an r-open cover of Z, say 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } , that cannot be reducible to a finite subcover of Z. Thus, we have: 𝑍 = ⋃ 𝛼 ∈ Λ 𝑢 𝛼 . (17) Consequently, 𝜙 = ( ⋃ 𝛼 ∈ Λ 𝑢 𝛼 ) ∁ = ⋂ 𝛼 ∈ Λ 𝑢 𝛼 ∁ , and because 𝑢 𝛼 is r-open ∀ 𝛼 ∈ Λ , 𝛼 ∈ Λ ∁ is r-closed ∀ 𝛼 ∈ Λ . Consequently, 𝐹 ∼ = 𝑢 𝛼 ∁ 𝛼 ∈ Λ is a family of r-closed subsets of Z. Now, to finish this proof, we should concern ourselves with the following claim: Claim: 𝐹 ∼ has the F.I.P.

To prove this claim, we suppose by contrary that the above statement does not hold. Then, ∃ 𝑢 1 , ⋯ , 𝑢 𝑛 for which ⋂ 𝑘 = 1 𝑛 𝑢 𝑘 ∁ = 𝜙 . Thus, 𝑍 = ⋃ 𝑘 = 1 𝑛 𝑢 𝑘 , and so 𝑈 ∼ has a finite subcover of Z, which is a contradiction. Therefore, 𝐹 ∼ = 𝑢 𝛼 ∁ : 𝛼 ∈ Λ has the F.I.P., and so by the assumption ⋂ 𝛼 ∈ Λ 𝐹 𝛼 ≠ 𝜙 , we get 𝜙 ≠ ⋂ 𝛼 ∈ Λ 𝑢 𝛼 ∁ , which implies 𝑍 ≠ 𝑍 − ( ⋂ 𝛼 ∈ Λ 𝑢 𝛼 ∁ ) . This means 𝑍 ≠ ⋃ 𝛼 ∈ Λ 𝑢 𝛼 , which contradicts (17). Hence, Z is an r-compact spaceterm. □ A bitopological spaceterm 𝑍 = ( 𝑍 , 𝛾 1 , 𝛾 2 ) is a pairwise r-extremely disconnected compact spaceterm if and only if every family of 𝛾 𝑖 -closed subsets of Z with the F.I.P. has a nonempty intersection, ∀ 𝑖 = 1 , 2 .

⇒ ) Assume that 𝑖 = 1 , 2 , and Z is a pairwise r-extremely disconnected compact spaceterm. Suppose that ∃ a family 𝐹 ∼ = { 𝐹 𝛼 : 𝛼 ∈ Λ } of 𝛾 𝑖 -r-closed subsets of Z with the F.I.P. and ⋂ 𝛼 ∈ Λ 𝐹 𝛼 = 𝜙 . This implies ⋃ 𝛼 ∈ Λ 𝐹 𝛼 ∁ = ( ⋂ 𝛼 ∈ Λ 𝐹 𝛼 ) ∁ = 𝑍 . However, 𝐹 𝛼 is 𝛾 𝑖 -r-closed in Z, ∀ 𝛼 ∈ Λ , and so 𝐹 𝛼 ∁ is 𝛾 𝑖 -r-open in Z, ∀ 𝛼 ∈ Λ . Consequently, we have 𝑉 ∼ = { 𝐹 𝛼 ∁ : 𝛼 ∈ Λ } , for a 𝛾 𝑖 -r-open cover of Z. Then, by the assumption 𝑍 = ⋃ 𝑘 = 1 𝑛 𝐹 𝛼 𝑘 ∁ = ( ⋂ 𝑘 = 1 𝑛 𝐹 𝛼 𝑘 ) ∁ , we can have 𝜙 = ⋂ 𝑘 = 1 𝑛 𝐹 𝛼 𝑘 , which is a contradiction with 𝐹 ∼ that has the F.I.P. Hence, the result holds.

⇐ ) Conversely, suppose every family of 𝛾 𝑖 -r-closed subsets of Z with the F.I.P. has a nonempty intersection. Assume that Z is not a pairwise r-compact spaceterm. Then, ∃ a 𝛾 𝑖 -r-open cover of Z, say 𝑈 ∼ = { 𝑢 𝛼 : 𝛼 ∈ Λ } that cannot be reducible to a finite subcover of Z, and hence we have: 𝑍 = ⋃ 𝛼 ∈ Λ 𝑢 𝛼 . (18) Consequently, we obtain 𝜙 = ( ⋃ 𝛼 ∈ Λ 𝑢 𝛼 ) ∁ = ⋂ 𝛼 ∈ Λ 𝑢 𝛼 ∁ . Since 𝑢 𝛼 is 𝛾 𝑖 -r-open ∀ 𝛼 ∈ Λ , then 𝛼 ∈ Λ ∁ is 𝛾 𝑖 -r-closed ∀ 𝛼 ∈ Λ . This leads to assert that 𝐹 ∼ = 𝑢 𝛼 ∁ 𝛼 ∈ Λ is a family of 𝛾 𝑖 -r-closed subsets of Z. Now, to finish this proof, we should concern with the following claim: Claim: 𝐹 ∼ has the F.I.P.

To prove this claim, we suppose on the contrary that the above statement does not hold. Then, there exist 𝑢 1 , ⋯ , 𝑢 𝑛 such that ⋂ 𝑘 = 1 𝑛 𝑢 𝑘 ∁ = 𝜙 . Then, 𝑍 = ⋃ 𝑘 = 1 𝑛 𝑢 𝑘 , and so 𝑈 ∼ has a finite subcover of Z, which is a contradiction. Thus, 𝐹 ∼ = 𝑢 𝛼 ∁ : 𝛼 ∈ Λ has the F.I.P., and so by the assumption ⋂ 𝛼 ∈ Λ 𝐹 𝛼 ≠ 𝜙 , we have 𝜙 ≠ ⋂ 𝛼 ∈ Λ 𝑢 𝛼 ∁ , which implies 𝑍 ≠ 𝑍 − ( ⋂ 𝛼 ∈ Λ 𝑢 𝛼 ∁ ) . This means 𝑍 ≠ ⋃ 𝛼 ∈ Λ 𝑢 𝛼 , which contradicts (18). Hence, Z is a pairwise r-compact spaceterm. □ 4. Conclusions In this work, the so-called pairwise r-compactness was well-defined in topological and bitopological spacesterm. Several properties of these spaces with their relations with other topological and bitopological spacesterm were consequently established theoretically. The inferred results can pave the way to deriving other novel theorems related to the finite product and mappings of pairwise expandable spaces, feebly pairwise expandable spaces and fuzzy bitopological spacesterm, which is left to future considerations.