Abstract

In this paper we define P_Σ- and co-P_Σ- ditopological texture spaces. Fundamental properties of these structures have been explored. Also we have put forth few new concepts regarding continuity of difunctions.

1. Introduction

In 1996, Khan et al. in ¹ studied P_Σ- topological spaces and weakly P_Σ- topological spaces. They observed that properties of P_Σ- topological spaces ascribed by Wang ² and strongly S-regular ascribed by Ganster ³ are analogous to one another. Let S be a topological space than it is called P_Σ- topological space if every open subset of the topological space S is expressed as the union of regular closed sets. S is said to be weakly P_Σ- topological space if every regular open subset of S is expressed as the union of regular closed sets.

During the last decade of 20th century, L. M. Brown ^{4, 5} gave the idea of texture under fuzzy structure. He proposed that idea at the second BUFSA Conference on Fuzzy Systems and Artificial Intelligence that occurred at Trabzon University, Turkey in 1992. In brief, let S be a set. We recall ^{4, 6, 7, 8} that a texturing δ on S is point-separating, complete and completely distributive lattice which contains S and φ and also for which meet coincides with intersection and finite join coincides with union. The pair is then known as a texture space. The marvels of texture spaces expand forward. When research is done on fuzzy topological structures, ditopological texture spaces are considered. Ditopologies are built upon a partially ordered set assuring the principles of texture. A ditopology naturally associates to a fuzzy topology as does the texture space. From ^{4, 5, 6, 7, 9, 10, 11, 12}, one can acquire a sufficient prologue to the hypothesis of texture spaces as well as ditopological texture spaces and the inspiration for its investigation. Analogue to the concepts given above, the concepts of P_Σ- topological spaces and ditopological texture spaces are amalgamated in this research paper.

2. Preliminaries

Definition 2.1. ⁵ Let us take a texture space (S, δ). A ditopology which is also known as dichotomous topology defined on a given texture is a pair (τ, κ) comprises of sets contained in texturing δ where τ contains the collection of open sets which fulfills the following properties:

(a)

(b) If then

(c) If where then

and κ contains the collection of closed sets which fulfills the following properties:

(d)

(e) If then

(f ) If where then

There is clearly no connection among open sets and closed sets in ditopology.

Definition 2.2. ¹³ Let us consider a ditopological texture space Then interior and closure respectively of the set can be described mathematically as and

Definition 2.3. ¹⁴ Let us consider a ditopology on a texture space Then an element is known to be regular open (RO) if and an element is known to be regular closed (RC) if

Definition 2.4. ⁹ Let us consider a direlation from to Then we say that is a difunction from to if it fulfills the accompanying properties:

DF1: For with and

DF2: For and and

Definition 2.5. ⁹ Let us consider a difunction

(a) For and respectively said to be the image and co-image are interpreted as follows:

and

(b) For and respectively said to be the inverse image and the inverse co-image are interpreted as follows:

and

Definition 2.6. ⁹ Let be a difunction from a texture space to the texture space Then is said to be surjective difunction if it fulfills the following property:

SUR: For with and

In a similar manner, is said to be injective if it fulfills the following property:

INJ: For and and

Definition 2.7. ¹³ Let us consider a ditopological texture space and a difunction from the texture space to the texture space Then:

(a) is continuous if for every

(b) is cocontinuous if for every

(c) is known to be bicontinuous if it continuous and cocontinuous too.

Definition 2.8. ¹³ Let us consider a ditopological texture space and a difunction from the texture space to the texture space Then:

(a) is known to be open (co-open) if:

(b) is known to be closed (co-closed) if:

Definition 2.9. ¹⁵ Let be a difunction. Then:

(a) is almost continuous if for every

(b) is almost co-continuous, if for every

Definition 2.10. ¹⁴ Let be a difunction. Then:

(a) is an R-dimap if for every

(b) is a co-R-dimap if for every

3. P_Σ and Co-P_Σ Ditopological Texture Spaces

Definition 3.1. Let be a ditopological texture space. Then is P_Σ- ditopological texture space if every open set can be expressed as join of regular closed sets.

Example 3.1. Let and is a ditopology on Consider Clearly every open set can be expressed as join of regular closed sets. So, is P_Σ- ditopological texture space.

Theorem 3.1. Let be a ditopological texture space. Then S is P_Σ- ditopological texture space if and only if for any open subset P of S and for any point an such that

Proof 3.1. Let be P_Σ- ditopological texture space. We prove that for any open subset P of S and for any point an such that Let This implies where Now for any point some such that This implies This gives where

Conversely, consider for any open subset P of S and for any point an such that We prove that S is P_Σ- ditopological texture space. Let such that C is open. Let Then by our supposition, some such that Thus,

This implies where Thus S is P_Σ- ditopological texture space.

Definition 3.2. Let be a ditopological texture space. Then is co-P_Σ- ditopological texture space if every closed set can be expressed as the intersection of regular open sets.

Example 3.2. Let and is a ditopology on Consider Clearly, every closed set can be expressed as the intersection of regular open sets. So, is co-P_Σ- ditopological texture space.

Theorem 3.2. Let be a ditopological texture space. Then S is co-P_Σ- ditopological texture space if for any closed subset P of S and for any point an such that

Proof 3.2. Let be co-P_Σ- ditopological texture space. We prove that for any closed subset P of S and for any point an such that Let This implies where Now for any point some such that This gives This implies where Hence proved.

4. Weakly P_Σ and Weakly Co-P_Σ Ditopological Texture Spaces

Definition 4.1. Let be a ditopological texture space. Then is weakly-P_Σ ditopological texture space if every regular open subset of S can be expressed as join of regular closed sets.

Example 4.1. Let and is a ditopology on Consider Further

As every regular open subset of S can be expressed as the join of regular closed sets. So, is weakly-P_Σ- ditopological texture space.

Theorem 4.1. Let be a ditopological texture space. Then S is weakly P_Σ- ditopological texture space if and only if for any regular open subset P of S and for any point an such that

Proof 4.1. Similar to the proof of Theorem 3.1.

Definition 4.2. Let be a ditopological texture space. Then is weakly co-P_Σ ditopological texture space if every regular closed subset of S can be expressed as intersection of regular open sets.

Example 4.2. Let and is a ditopology on Consider Further, As every regular closed subset of S can be expressed as the intersection of regular open sets. So, is weakly co-P_Σ- ditopological texture space.

Theorem 4.2. Let be a ditopological texture space. Then S is weakly co-P_Σ- ditopological texture space if for any regular closed subset P of S and for any point an such that

Proof 4.2. Similar to the proof of Theorem 3.2.

5. Continuity of Difunction

Definition 5.1. Let be a difunction. Then:

(a) is completely continuous if for every

(b) is completely co-continuous if for every

(c) is almost open (almost co-open) if for every

(d) is almost closed (almost co-closed) if for every

(e) is pre-almost open (pre-almost co-open) if for every

(f) is pre-almost closed (pre-almost co-closed) if for every

(g) is regular open (regular co-open) if for every

(h) is regular closed (regular co-closed), if for every

Lemma 5.1. Let us consider a difunction

(a) If is pre-almost open (pre-almost co-open) then it is almost open (almost co-open).

(b) If is pre-almost closed (pre-almost co-closed) then it is almost closed ( almost co-closed).

(c) If is regular open (regular co-open) then it is pre-almost open (pre-almost co-open).

(d) If is regular closed (regular co-closed) then it is pre-almost closed (pre-almost co-closed).

(e) If is completely continuous (completely co-continuous) then it is an R-dimap (co-R-dimap).

Theorem 5.1. If be a co-R-dimap and open injection and is P_Σ- ditopological texture space, then is P_Σ- ditopological texture space.

Proof 5.1. Let Q be an arbitrary open subset of then is open in As is P_Σ- ditopological texture space and hence where Since is injective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is a co-R-dimap and hence we have by using [ ⁹, Theorem 2.24]. Therefore, is P_Σ- ditopological texture space.

Theorem 5.2. If be an R-dimap and co-closed injection and is co-P_Σ- ditopological texture space, then is co-P_Σ- ditopological texture space.

Proof 5.2. Similar to the proof of Theorem 5.1.

Theorem 5.3. If be a co-R-dimap and almost open injection and is P_Σ- ditopological texture space, then is weakly P_Σ- ditopological texture space.

Proof 5.3. Let Q be an arbitrary regular open subset of then is open in As is P_Σ- ditopological texture space and hence where Since is injective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is a co-R-dimap and hence we have by using [ ⁹, Theorem 2.24]. Therefore, is weakly P_Σ- ditopological texture space.

Theorem 5.4. If be an R-dimap and almost co-closed injection and is co-P_Σ- ditopological texture space, then is weakly co-P_Σ- ditopological texture space.

Proof 5.4. Similar to the proof of Theorem 5.3.

Theorem 5.5. If be a co-R-dimap and pre-almost open injection and is weakly P_Σ- ditopological texture space, then is weakly P_Σ- ditopological texture space.

Proof 5.5. Let Q be an arbitrary regular open subset of then is regular open in As is weakly P_Σ- ditopological texture space and hence where Since is injective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is a co-R-dimap and hence we have by using [ ⁹, Theorem 2.24]. Therefore, is weakly P_Σ- ditopological texture space.

Theorem 5.6. If be an R-dimap and pre-almost co-closed injection and is weakly co-P_Σ- ditopological texture space, then is weakly co-P_Σ- ditopological texture space.

Proof 5.6. Similar to the proof of Theorem 5.5.

Theorem 5.7. If be a co-R-dimap and regular open injection and is weakly P_Σ- ditopological texture space, then is P_Σ- ditopological texture space.

Proof 5.7. Let Q be an arbitrary open subset of then is regular open in As is weakly P_Σ- ditopological texture space and hence where Since is injective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is a co-R-dimap and hence we have by using [ ⁹, Theorem 2.24]. Therefore, is P_Σ- ditopological texture space.

Theorem 5.8. If be an R-dimap and regular co-closed injection and is weakly co-P_Σ- ditopological texture space, then is co-P_Σ- ditopological texture space.

Proof 5.8. Similar to the proof of Theorem 5.7.

Theorem 5.9. If be continuous and pre-almost closed surjection and is P_Σ- ditopological texture space, then is P_Σ- ditopological texture space.

Proof 5.9. Let Q be an arbitrary open subset of then is open in As is P_Σ- ditopological texture space hence where As is surjective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is pre-almost closed and hence we have This shows that is P_Σ- ditopological texture space.

Theorem 5.10. If be co-continuous and pre-almost co-open surjection and is co-P_Σ- ditopological texture space, then is co-P_Σ- ditopological texture space.

Proof 5.10. Similar to the proof of Theorem 5.9.

Theorem 5.11. If be completely continuous and pre-almost closed surjection and is weakly P_Σ- ditopological texture space, then is P_Σ- ditopological texture space.

Proof 5.11. Let Q be an arbitrary open subset of then is regular open in As is weakly P_Σ- ditopological texture space hence where As is surjective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is pre-almost closed and hence we have This shows that is P_Σ- ditopological texture space.

Theorem 5.12. If be completely co-continuous and pre-almost co-open surjection and is weakly co-P_Σ- ditopological texture space, then is co-P_Σ- ditopological texture space.

Proof 5.12. Similar to the proof of Theorem 5.11.

Theorem 5.13. If be an R-dimap and pre-almost closed surjection and is weakly P_Σ- ditopological texture space, then is weakly P_Σ- ditopological texture space.

Proof 5.13. Let Q be an arbitrary regular open subset of then is regular open in As is weakly P_Σ- ditopological texture space hence where As is surjective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]

Also it is given that is pre-almost closed and hence we have This shows that is weakly P_Σ- ditopological texture space.

Theorem 5.14. If be a co-R-dimap and pre-almost co-open surjection and is weakly co-P_Σ- ditopological texture space, then is weakly co-P_Σ- ditopological texture space.

Proof 5.14. Similar to the proof of Theorem 5.13.

Theorem 5.15. If be almost continuous and pre-almost closed surjection and is P_Σ- ditopological texture space, then is weakly P_Σ- ditopological texture space.

Proof 5.15. Let Q be an arbitrary regular open subset of then is open in As is P_Σ- ditopological texture space hence where As is surjective, so by [ ⁹, Corollary 2.12 and Corollary 2.33]:

Also it is given that is pre-almost closed and hence we have This shows that is weakly P_Σ- ditopological texture space.

Theorem 5.16. If be almost co-continuous and pre-almost co-open surjection and is co-P_Σ- ditopological texture space, then is weakly co-P_Σ- ditopological texture space.

Proof 5.16. Similar to the proof of Theorem 5.15.

References