Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces
Abstract
:1. Introduction
- (i)
- is called if it is closed under the formation of finite suprema and if it is closed under the operation of taking smaller function.
- (ii)
- dominates d if and there exists such that and if dominates d, then is called saturated.
- (i)
- To explicitly characterize each of , and separation properties in the category of quantale-valued gauge spaces and -gauge morphisms;
- (ii)
- To give the characterization of each of Pre-, and in the category ;
- (iii)
- To examine the mutual relationship among all these separation axioms;
- (iv)
- To compare our results with the ones in some other categories.
2. Preliminaries
- (i)
- is a semi group;
- (ii)
- and for all and index-set I, i.e., ∗ is distributive over arbitrary joins.
- (i)
- if such that there exists such that ;
- (ii)
- if such that there exists such that .
- (i)
- a commutative quantale if is a commutative semi-group;
- (ii)
- an integral quantale if for all ;
- (iii)
- a value quantale if is commutative and integral quantale with an underlying completely distributive lattice such that and for all ;
- (iv)
- a linearly ordered quantale if either or for all .
- (i)
- If , then an -metric space is a preordered set.
- (ii)
- If is a Lawvere’s quantale, then an -metric space is an extended pseudo-quasi metric space.
- (iii)
- If , then an -metric space is a probabilistic quasi metric space [23].
- (i)
- m is called locally supported by if for all , , , there is such that .
- (ii)
- is called locally directed if for all finite subsets , is locally supported by .
- (iii)
- is called locally saturated if for all we have whenever m is locally supported by .
- (iv)
- The set is locally supported by is called local saturation of .
- (i)
- .
- (ii)
- and implies .
- (iii)
- implies .
- (iv)
- is locally saturated.
3. and Quantale-Valued Approach Spaces
4. (Pre-)Hausdorff -Gauge Spaces
- (I)
- For all with , .
- (II)
- For any three distinct points , .
- (III)
- For any four distinct points , .
- (I)
- (II)
- or or
- (III)
- or oror oror
- (I)
- For all with , .
- (II)
- For any three distinct points , .
- (III)
- For any four distinct points , .
- (I)
- (II)
- or or
- (III)
- or oror oror
- 1.
- is
- 2.
- is
- 3.
- is
5. Comparative Evaluation
- 1.
- -
- 2.
- -
- 3.
- -
- 4.
- 5.
- (1)
- In ,
- (a)
- By Theorems 1–3 and 5, .
- (b)
- By Theorems 4–6, if an -gauge space is , then is both and Pre-.
- (c)
- By Theorem 7, is a Pre-Hausdorff -gauge space, then , and are equivalent.
- (2)
- (3)
- (4)
- (a)
- (b)
- (c)
- (d)
- (e)
- (5)
- (a)
- (b)
6. Conclusions
- (i)
- Can one characterize each of , , irreducible, compact, connected, sober and zero-dimensional quantale-valued gauge spaces?
- (ii)
- Can one present the Urysohn’s Lemma, the Tietze Extension Theorem and the Tychonoff Theorem for the category ?
- (iii)
- How can one characterize , , , Pre-, and separation axioms for quantale generalization of other approach structures such as approach distances and approach systems, and what would be their relation to each other?
- (iv)
- In the category of approach spaces and contraction maps, what would be the characterization of Pre-, and properties?
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Özkan, S.; Alsulami, S.; Baran, T.M.; Qasim, M. Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces. Mathematics 2022, 10, 4819. https://doi.org/10.3390/math10244819
Özkan S, Alsulami S, Baran TM, Qasim M. Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces. Mathematics. 2022; 10(24):4819. https://doi.org/10.3390/math10244819
Chicago/Turabian StyleÖzkan, Samed, Samirah Alsulami, Tesnim Meryem Baran, and Muhammad Qasim. 2022. "Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces" Mathematics 10, no. 24: 4819. https://doi.org/10.3390/math10244819
APA StyleÖzkan, S., Alsulami, S., Baran, T. M., & Qasim, M. (2022). Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces. Mathematics, 10(24), 4819. https://doi.org/10.3390/math10244819