p-Regularity and p-Regular Modification in ⊤-Convergence Spaces
Abstract
:1. Introduction
2. Preliminaries
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- ;
- (5)
- ;
- (6)
- .
⊤-Filters and ⊤-Convergence Spaces
- (TF1)
- ,
- (TF2)
- , ,
- (TF3)
- if , then .
- (TB1)
- , ,
- (TB2)
- if , then .
- (1)
- For any , the family forms a ⊤-filter base on Y. The generated ⊤-filter is denoted as , called the image of under f. It is known that .
- (2)
- For any , the family forms a ⊤-filter base on Y iff holds for all . The generated ⊤-filter (if exists) is denoted as , called the inverse image of under f. It is known that holds whenever exists. Furthermore, always exists and whenever f is surjective.
- (3)
- For any , the family is a ⊤-filter on X, and .
- (1)
- If is a ⊤-filter base of , then is a ⊤-filter base of , see Example 2.9 (1) in [31].
- (2)
- If is a ⊤-filter base of and exists, then is a ⊤-filter base of , see Example 2.9 (2) in [31].
- (3)
- Let and be a ⊤-filter base of . Then implies that , see Lemma 2.5 (1) in [42].
- (4)
- Let and be a ⊤-filter base of . Then , see Lemma 3.1 in [36].
- (TC1)
- , ;
- (TC2)
- if and , then .
3. p-Regularity in ⊤-Convergence Spaces
- (1)
- ;
- (2)
- implies ;
- (3)
- .
- (1)
- ,
- (2)
- if , then ,
- (3)
- if and , then .
- (1)
- If f is a continuous function, then .
- (2)
- If f is a closure function, then .
4. Lower (Upper) p-Regular Modifications in ⊤-Convergence Spaces
- (1)
- If q is p-regular, then implies for any .
- (2)
- If q is p-regular, then q is -regular for any .
- (3)
- The indiscrete structure ι is p-regular for any .
4.1. Lower p-Regular Modification
4.2. Upper p-Regular Modification
- (1)
- . It is obvious.
- (2)
- . In fact, let then .
- (3)
- is p-regular. In fact, let . Then for any it holds that which means . So, is p-regular.
- (4)
- Let r be p-regular with . Then . In fact, let then for any , by Proposition 6 (1) it holds that and so by . That means .
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Jin, Q.; Li, L.; Lang, G. p-Regularity and p-Regular Modification in ⊤-Convergence Spaces. Mathematics 2019, 7, 370. https://doi.org/10.3390/math7040370
Jin Q, Li L, Lang G. p-Regularity and p-Regular Modification in ⊤-Convergence Spaces. Mathematics. 2019; 7(4):370. https://doi.org/10.3390/math7040370
Chicago/Turabian StyleJin, Qiu, Lingqiang Li, and Guangming Lang. 2019. "p-Regularity and p-Regular Modification in ⊤-Convergence Spaces" Mathematics 7, no. 4: 370. https://doi.org/10.3390/math7040370
APA StyleJin, Q., Li, L., & Lang, G. (2019). p-Regularity and p-Regular Modification in ⊤-Convergence Spaces. Mathematics, 7(4), 370. https://doi.org/10.3390/math7040370