Life Science Journal 2014;11(9) http://www.lifesciencesite.com

Cardinal Functions on Ditopological Texture Spaces

Kadirhan POLAT1, Ceren Sultan ELMALI2, Tamer UĞUR1

1Mathematics Department, Faculty of Science, Ataturk University, 25240, Erzurum, Turkey 2Mathematics Department, Faculty of Science, Erzurum Techinal University, 25240, Erzurum, Turkey [email protected], [email protected], [email protected]

Abstract: In this paper we define the concept of dicardinal , and then (co)weight, (co)densification, (co)net weight, (co)pseudo character which are able to be used in classifying of ditopological texture spaces. It is natural to ask how there are relationships between the ( or ) and dicardinal functions that we defined in ditopological texture spaces. Based on the question, our aim in the paper is to investigate dicardinal functions above for ditopological texture spaces. We obtain useful some results on bounds of , the set of all -sets and the set of all -sets by choosing the subclasses satisfying axiom or Co- the class of all ditopological texture spaces. Furthermore, we show that (co)weight and (co)densification restrict each others. [Polat K, Elmalı CS, Uğur T. Cardinal Functions on Ditopological Texture Spaces. Life Sci J 2014;11(9):293-297]. (ISSN:1097-8135). http://www.lifesciencesite.com. 40

Keywords: Texture space; ditopology; ditopological texture space; cardinal function; cardinal invariant; dicardinal function; dicardinal invariant MSC: 54A25; 54A40

1. Introduction space, it is obvius intiutively that a convenient Being closest to set theory, cardinal on a texture space doesn’t need to hold the invariants play a major role in of existence of the duality of interior and closure and so which set theory forms the . They are used as a not need to hold both axioms of open sets and ones of most useful tools in classifying topological spaces. closed sets. In a series of three papers, L. M. Brown Some important classes of topological spaces et al. present first two ones in 2004 and last one in distinguished by them are separable spaces, compact 2006. In the first of them subtitled ’Basic concepts’, spaces, topological spaces which has a countable the authors introduce a systematic form of the basis. Also, by using cardinal invariants, it is possible concepts of direlation, difunction, the category dfTex to compare quantitively topological properties, and ditopological texture space in a categorical setting. In generalize the present results of them. Many second paper, the category dfDitop of ditopological researchers have contributed to development in texture spaces and bicontinuous difunctions is theory of cardinal functions since 1920. In the defined. The subject of third paper is on separation 1920’s, Alexandroff and Urysohn (1929) show that axioms in general ditopological texture space. In a every compact, perfectly normal space has ditopological setting , L. M. Brown and M. M. Gohar ≤ 2. One of results that Čech and Pospíšil (1938) study compactness in 2009, and strong compactness obtained states that every compact, first countable one year later [9-10]. space has cardinality ≤ or ≥ 2. In 1940’s, it is In this study, after given concept of shown by Pondiczery (1944), Hewitt (1946) and dicardinal function, (co)weight, (co) net weight, (co) Marczewski (1947) that a product of at most 2 densification, (co) pseudo character which are ones of is separable. In 1965, one of Groot’s most useful tools in classifying ditopological texture results which generalizes Alexandroff and Urysohn’s spaces are defined. It is natural to ask how there are result above states that a Hausdorff space in which relationships between the sets ( or ) and every subspace is Lindelöf has cardinality ≤ 2 . dicardinal functions which will be defined on Arhangel'skii (1969) show that every Lindelöf, first ditopological texture spaces. Based on the question, countable, Hausdorff space has cardinality ≤ 2. our aim in the paper is to investigate (co) weight, (co) The notion of fuzzy structure introduced by net weight, (co) pseudo character, co (densification) L. M. Brown (1993) in [1-2] is redenominated in for ditopological texture spaces. In the section 2 titled texture space developed one of this structure by L. M. ’Texture Spaces’, we recall the basic definitions of Brown and R. Ertürk (2000) in [4-5]. The structure texture space, ditopology on the texture space and makes it possible to be investigated mathematical then, some definitions and theorems regarding the concepts without any complement in account of the subjects. The concepts of ordinals, cardinals, fact that, in a texture space(, ), doesn’t need to cardinality of a set and cardinal functions and some be a closed set under set-theoretical complement by theorems which are related to cardinals are given in the definition above. Based on the structure of texture the section 3. Finally, in the last section, we give the

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definition of dicardinal function in ditopological texture spaces. Then, we represent dicardinal Dense and Codense: A set ∈ is said to be functions and some important theorems for (a dense (codense) in (, ) if [] = (][= ∅). particular subclass of) the class of all ditopological texture spaces. For every ditopological texture space Axiom: Let = (, , , ) be a ditopological = (, , , ) , -() ≤ () and () ≤ space, ( ∪ )∨ denote the set of arbitrary joins of sets in ( ∪ ) and ( ∪ )∩ the set of arbitrary -(); if is , then || ≤ max{(), ()} | | (),-() intersections of sets in ∪ . is said to be if and ≤ 2 . In particular, for ∨ every Kolmogorov ditopological coseparated texture (∀, ∈ )(∃ ∈ ( ∪ ) ) ( ⊈ ⇒ ⊈ ⊈ ), space, || ≤ max{(), ()}. If is - , then or equivalently, || ≤ ()(). (∀, ∈ )(∃ ∈ ( ∪ )∩) ( ⊈ ⇒ ⊈ ⊈ ). 2. Texture Spaces The following definitions and propositions Proposition 2.1. The following are characteristic were introduced in [1-11]. properties of ditopological texture space: A texturing on a non-empty set is a set (∀, ∈ )(∃ ∈ ∪ ) ( ⊈ ⇒ ⊈ ⊈ ), containing , ∅ of subsets of with respect to (∀, ∈ )([ ] ⊆ [ ] and ] [⊆] [⇒ ⊆ ). inclusion satisfying the conditions: () (, ⊆) is a complete lattice () is completely distributive, and - Axioms Let = (, , , ) be a () separates the points of , () Meets ⋀ ditopological space. is said to be if and finite joins ⋁ coincide with intersections ⋂ and (∀ ∈ )(∀ ∈ ) ( ⊈ ⇒ [ ] ⊈ ). unions ⋃, respectively, for . (, ) is then called texture space. is said to be - if For each ∈ , the -set is defined by (∀ ∈ )(∀ ∈ ) ( ⊈ ⇒ ⊆][. = ⋂{ ∈ | ∈ }, and -set and - Axioms Let = (, , , ) be a = ⋁{ ∈ | ∉ } = ⋁{ | ∈ , ∉ }. ditopological space. is said to be - if it We recall that a texture (, ) is said to be is and -. coseparated if, for all , ∈ , ⊆ ⇒ ⊆ . Proposition 2.2. The following are characteristic We define = { | ∈ } and = { | ∈ }. properties of ditopological texture space: (∀ ∈ )(∃ℱ ⊆ ) ( = ⋁ℱ), Texture space: Let (, ) a texture space, and , (∀, ∈ )(∃ ∈ ) ( ⊈ ⇒ ⊈ ⊈ ). subsets of . = (, , , ) is called a ditopological texture space, and the pair (, ) a Proposition 2.3. The following are characteristic ditopology on (, ) if (, ) satisfies properties of - ditopological texture space: (1) , ∅ ∈ , (∀ ∈ )(∃ ⊆ ) ( = ⋂), (2) , ∈ ⇒ ∩ ∈ , (∀, ∈ )(∃ ∈ ) ( ⊈ ⇒ ⊈ ⊈ ). (3) ⊆ ⇒ ⋁ ∈ , (1) , ∅ ∈ , 3. Cardinal Functions (2) , ∈ ⇒ ∪ ∈ , The following definitions and propositions (3) ℱ ⊆ ⇒ ⋂ℱ ∈ . were introduced in [12-24].

Base and Cobase: Let = (, , , ) a Cardinal and Ordinal: A set is called transitive ditopological texture space, and ℬ a subset of (). iff ∀∀(∈∧∈⇒∈). An ordinal is ℬ is a (cobase) for (, ) if, for all ∈ (), a transitive set such that all ∈ are transitive. there exists a subset ℬ of ℬ such that = Then, we recall that a cardinal is an ordinal that there ⋁ℬ (⋂ℬ). is no bijection from itself to a smaller ordinal.

Interior and Closure: Let = (, , , ) be a Proposition 3.1. If (, ≺) is a well-ordering, then ditopological texture space. The interior and the there is unique ordinal and a unique isomorphism closure of the set ∈ is defined, respectively: ∶ (, ≺) → (, ∈). ][= ⋁{ | ⊆ }, [] = ⋂{ | ⊆ }. Proposition 3.2. Assuming axiom of choice, for

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every set , there exists a relation ≺ on such numbers is called a dicardinal function if, for every that (, ≺) is a well-ordering. pair = ( , , , )∈{,} of ditopological texture spaces, Cardinality of a Set: Let be a set. From the facts ‘, are dihomeomorphic’ ⇒ () = (). above, there exists an ordinal that can be mapped one-to-one onto . The smallest one of the ordinals Number of Open and Closed Sets: Let = is called the cardinality or of , (, , , ) be a ditopological texture space. () written as ||. and () are defined as the number of open sets in plus and the number of closed sets in plus Sum and Product of Cardinals: Let and be , respectively. () is defined as the number of cardinal numbers. The cardinal number + ≔ sets in ∪ plus . | × {0} ∪ × {1}| is called the cardinal sum of and , and . ≔ | × | cardinal product of Remark 4.1. Clearly, max{(), ()} ≤ () ≤ and . || ≤ 2|| . Also, neither of () and () dominates the other according to [?, Section 2]. If Proposition 3.3. If one of cardinal numbers and is a complemented ditopological texture space, then is nonzero and the other infinite, then + = . = () = (). max{, }. Let be a set. The set of all subsets of which Theorem 4.2. If a ditopological texture space have cardinality is denoted by [] , that is, = (, , , ) is (Kolmogorov), then [] = { ⊆ | || = } . The sets [] and || ≤ max{(), ()}. [] are defined analogously, that is, [] = { ⊆ | || < } and [] = { ⊆ | || ≤ }. Proof. Define ∶ → × by () = (][, []) . Since is , is one-one (See Proposition 3.4. Let an infinite set of cardinality Proposition 2.1.(2)). Thus || ≤ | × | = , and ≤ a cardinal. Then [] = . max{(), ()}. We recall that a cardinal function is a function Conclusion 4.1. For every Kolmogorov ditopological from the class of all topological spaces into the class coseparated texture space, || ≤ max{(), ()}. of all infinite cardinals such that, if and are homeomorphic, then () = (). Weight and Coweight: Let = (, , , ) be a ditopological texture space. The weight and coweight 4. Main Results of are defined as follows, Since separates the points of , for every () = min{|ℬ| | ℬ a base for (, )} , -() = min{|ℬ| | ℬ a cobase for (, )}, ,∈,≠⇒ ≠ . From the definition -set, respectively. for every , ∈ , ≠ ⇒ ≠ . Thus || = ||. It need to be true that || = ||. To prove the equality, we need to use the definition of coseparated Densifier and Codensifier: Let = (, , , ) be texture space. a ditopological texture space. A subset of is said to be densifier (codensifier) in if ⋁ (⋂) Theorem 4.1. If a texture space (, ) is is dense (codense) in (, ). coseparated, then || = ||. Densification and Codensification: The Proof. Consider a pair of distinct points , ∈ . densification and codensification of are defined as follows, Then ≠ , and so ⊈ or ⊈ . Since () = min{|| | densiier in (, )} , (, ) is coseparated, ⊈ or ⊈ ; -() = min{|| | codensiier in (, )}, therefore ≠ . Thusly, we have an one-one map from to . Now, take , ∈ with ≠ . respectively. Now, we show that, in a ditopological Then ⊈ or ⊈ . From the definition of -set, it is immadiate that ⊈ or ⊈ , and so texture space , how there are relationships between (co)weight and (co)densification. ≠ ; therefore ≠ . Thereby, we have an one-one map from to . Theorem 4.3. For every ditopological texture space Dicardinal Function: A function from the class = (, , , ), we have of all ditopological texture spaces (or a particular -() ≤ (), subclass) into the class of all infinite cardinal () ≤ -().

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2|×ℳ| ≤ 2(),-(). Proof. 1. Let ℬ = be a base for the ∈ ditopological texture space not consisting of ∅ Pseudo Base and Copseudo Base: Let = (, , , ) be a ditopological texture space. A subset such that |ℬ| = (). Let us set ℳ = ∈ of is called a pseudo base of for (, ) if by choosing a certain -set such that ⊈ = ⋂ for some ∈ . A subset of is for each ∈ . Since ℬ is a base for , called a copseudo base of for (, ) if = ⋂ ∀ ∈ , ≠ ∅ ⇒ ⊈ ⋂ℳ; for some ∈ . so ℬ is codensifier in . Let us define ∶ ℬ → ℳ by ℬ = . Obviously, is onto; therefore Pseudo Character and Copseudo Character: The pseudo character and copseudo character of are |ℳ| ≤ |ℬ| . Since -() ≤ |ℳ| , -() ≤ defined as follows, (). Ψ() = sup{Ψ(, ) | ∈ } , 2. Let ℬ = {}∈ be a cobase for the ditopological -Ψ() = sup{-Ψ(, ) | ∈ }, texture space not consisting of such that respectively, where |ℬ| = -() . Let us set = by ∈ Ψ(, ) = min{|| | a pseudo base of for (, )}, choosing a certain -set such that ⊈ for each ∈ . Since ℬ is a cobase for , -Ψ(, ) ∀ ∈ , ≠ ⇒ ⋁ ⊈ ; = min{|| | a copseudo base of for (, )}. so ℳ is densifier in . Let us define ∶ ℬ → Theorem 4.5. If a ditopological texture space by () = . Obviously, is onto; therefore () || ≤ |ℬ|. Since () ≤ ||, () ≤ -(). = (, , , ) is -, then || ≤ () .

Net and Conet: Let = (, , , ) be a Proof. Let ⊆ be a net for (, ) with || ≤ (), let be a pseudo base for (, ) ditopological texture space. A subset of is ∗ ( ) called a net for (, ) if there exists a subset of with ≤ Ψ . Set = | ∈ by ∗ such that = ⋁ for all ∈ . A subset ℳ choosing a certain element ∈ such that ( ) of is called a conet for , if there exists a ⊆ ⊆ for each ∈ . Then, it is clear that subset ℳ∗ of ℳ such that = ⋃ℳ∗ for all ( ) ≤ Ψ . Moreover, since is - , by ∈ . Preposition 2.3.(1), ⊆ ⋂ ⊆ ⋂ = ; thus ⋂ = . Then, the function Ψ ∶ → []() Net Weight and Conet Weight: The net weight and defined by the equation Ψ( ) = is one-one. conet weight of are defined as follows, ( ) () = min{|| | a net for (, )} , Thus || = || ≤ () . -() = min{|| | a conet for (, )}, respectively. Corresponding Author: Res. Assist. Kadirhan Polat Then, we show that net weight and conet Mathematics Department, weight have bounds on in . Faculty of Science, Ataturk University, 25240, Erzurum, Turkey Theorem 4.4. If a ditopological texture space E-mail: [email protected]

= (, , , ) is , then || ≤ 2(),-(). References 1. Brown LM. Ditopological fuzzy structures i. Fuzzy systems and A.I. Magazine 1993; 3 (1). Proof. Let ⊆ be a net for (, ) with 2. Brown LM. Ditopological fuzzy structures ii. || ≤ (), let ℳ ⊆ be a conet for (, ) Fuzzy systems and A.I. Magazine 1993; 3 (2). | | with ℳ ≤ -() . Since is (See 3. Brown LM, Diker M. Ditopological texture Proposition 2.1.(1)) and , ℳ a net and conet for spaces and intuitionistic sets. Fuzzy Sets and (, ), respectively, for all , ∈ , there exist a Systems 1998; 98 (2) : 217 – 224. member of and a member of of ℳ satisfying 4. Brown LM, Ertürk R. Fuzzy sets as texture ⊈ ⇒ ( ⊈ ⊈ ∨ ⊈ ℳ ⊈ ). spaces, i. representation theorems. Fuzzy Sets Set ℒ = {(, ) ∈ × ℳ | , ⊈ } for each and Systems 2000; 110 (2) : 227 – 235. ∈ . Define ∶ → ( × ℳ) by () = 5. Brown LM, Ertürk R. Fuzzy sets as texture | | ℒ . Then, clearly, is one-one. Thus ≤ spaces, ii. subtextures and quotient textures.

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