scholarly journals Fuzzy Counterparts of Fischer Diagonal Condition in ⊤-Convergence Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 685
Author(s):  
Qiu Jin ◽  
Lingqiang Li ◽  
Jing Jiang
Keyword(s):  
Topological Space ◽  
Fuzzy Topology ◽  
Convergence Space ◽  
Convergence Spaces ◽  
Different Types ◽  
Diagonal Condition ◽  
Operator Convergence

Fischer diagonal condition plays an important role in convergence space since it precisely ensures a convergence space to be a topological space. Generally, Fischer diagonal condition can be represented equivalently both by Kowalsky compression operator and Gähler compression operator. ⊤-convergence spaces are fundamental fuzzy extensions of convergence spaces. Quite recently, by extending Gähler compression operator to fuzzy case, Fang and Yue proposed a fuzzy counterpart of Fischer diagonal condition, and proved that ⊤-convergence space with their Fischer diagonal condition just characterizes strong L-topology—a type of fuzzy topology. In this paper, by extending the Kowalsky compression operator, we present a fuzzy counterpart of Fischer diagonal condition, and verify that a ⊤-convergence space with our Fischer diagonal condition precisely characterizes topological generated L-topology—a type of fuzzy topology. Hence, although the crisp Fischer diagonal conditions based on the Kowalsky compression operator and the on Gähler compression operator are equivalent, their fuzzy counterparts are not equivalent since they describe different types of fuzzy topologies. This indicates that the fuzzy topology (convergence) is more complex and varied than the crisp topology (convergence).

scholarly journals A better framework for first countable spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 289
Author(s):  
Gerhard Preuss
Keyword(s):  
Topological Space ◽  
Uniform Space ◽  
Topological Spaces ◽  
Convergence Space ◽  
Convergence Spaces ◽  
Natural Function ◽  
Topological Construct ◽  
Countable Space ◽  
Filter Space ◽  
First Countable Space

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

scholarly journals More on Fuzzy Topological Spaces on Fuzzy Space

2021 ◽  
Vol 8 ◽  
pp. 21-26
Author(s):  
Abd Ulazeez Alkouri ◽  
Mohammad Hazaimeh ◽  
Ibrahim Jawarneh
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Fuzzy Topology ◽  
Separation Axioms ◽  
Fuzzy Space ◽  
Different Types ◽  
Fuzzy Topological Space ◽  
Isolated Points ◽  
Fuzzy Topological Spaces ◽  
Fuzzy Subsets

The fuzzy topological space was introduced by Dip in 1999 depending on the notion of fuzzy spaces. Dip’s approach helps to rectify the deviation in some definitions of fuzzy subsets in fuzzy topological spaces. In this paper, further definitions, and theorems on fuzzy topological space fill the lack in Dip’s article. Different types of fuzzy topological space on fuzzy space are presented such as co-finite, co-countable, right and left ray, and usual fuzzy topology. Furthermore, boundary, exterior, and isolated points of fuzzy sets are investigated and illustrated based on fuzzy spaces. Finally, separation axioms are studied on fuzzy spaces

scholarly journals A Tychonoff theorem in intuitionistic fuzzy topological spaces

2004 ◽  
Vol 2004 (70) ◽  
pp. 3829-3837
Author(s):  
Doğan Çoker ◽  
A. Haydar Eş ◽  
Necla Turanli
Keyword(s):  
Topological Space ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Topological Spaces ◽  
Fuzzy Topology ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Tychonoff Theorem

The purpose of this paper is to prove a Tychonoff theorem in the so-called “intuitionistic fuzzy topological spaces.” After giving the fundamental definitions, such as the definitions of intuitionistic fuzzy set, intuitionistic fuzzy topology, intuitionistic fuzzy topological space, fuzzy continuity, fuzzy compactness, and fuzzy dicompactness, we obtain several preservation properties and some characterizations concerning fuzzy compactness. Lastly we give a Tychonoff-like theorem.

scholarly journals Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1072 ◽  
Author(s):  
Sang-Eon Han ◽  
Saeid Jafari ◽  
Jeong Kang
Keyword(s):  
Applied Sciences ◽  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Discrete Topology ◽  
Separation Axiom ◽  
Different Types

The present paper deals with two types of topologies on the set of integers, Z : a quasi-discrete topology and a topology satisfying the T 1 2 -separation axiom. Furthermore, for each n ∈ N , we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on Z n , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.

Regular completions of uniform convergence spaces

1974 ◽  
Vol 11 (3) ◽  
pp. 413-424 ◽  
Author(s):  
R.J. Gazik ◽  
D.C. Kent ◽  
G.D. Richardson
Keyword(s):  
Uniform Convergence ◽  
Real Line ◽  
Dual Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Convergence Space ◽  
Uniform Spaces ◽  
Convergence Spaces ◽  
Uniform Convergence Space ◽  
Uniformly Continuous

A regular completion with universal property is obtained for each member of the class of u–embedded uniform convergence spaces, a class which includes the Hausdorff uniform spaces. This completion is obtained by embedding each u-embedded uniform convergence space (X, I) into the dual space of a complete function algebra composed of the uniformly continuous functions from (X, I) into the real line.

scholarly journals Applications of convergence spaces

1980 ◽  
Vol 21 (1) ◽  
pp. 107-123 ◽  
Author(s):  
Gary D. Richardson
Keyword(s):  
Convergence Space ◽  
Convergence Spaces ◽  
Space Setting ◽  
Convergence Structure ◽  
Probability And Statistics ◽  
Compactness Arguments

Convergence notions are used extensively in the areas of probability and statistics. Many times proofs can be simplified by considering an appropriate convergence structure on the space and using well-known results from the theory of convergence spaces; for example, compactness arguments are sometimes simplified by using a generalized Ascoli theorem in the convergence space setting. The theory of convergence spaces is also used to generalize some results in probability and statistics.

scholarly journals Quasi-uniform and uniform convergence of Riemann and Riemann-type integrable functions with values in a Banach space

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1899-1913
Author(s):  
Pratikshan Mondal ◽  
Lakshmi Dey ◽  
Ali Jaker
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Uniform Convergence ◽  
Sequence Space ◽  
Metric Space ◽  
Uniform Limit ◽  
Bounded Interval ◽  
Integrable Functions ◽  
Different Types ◽  
Sequence Of Functions

In this article, we study quasi-uniform and uniform convergence of nets and sequences of different types of functions defined on a topological space, in particular, on a closed bounded interval of R, with values in a metric space and in some cases in a Banach space. We show that boundedness and continuity are inherited to the quasi-uniform limit, and integrability is inherited to the uniform limit of a net of functions. Given a sequence of functions, we construct functions with values in a sequence space and consequently we infer some important properties of such functions. Finally, we study convergence of partially equi-regulated* nets of functions which is shown to be a generalized notion of exhaustiveness.

scholarly journals Maximal point spaces of posets with relative lower topology

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2645-2661
Author(s):  
Chong Shen ◽  
Xiaoyong Xi ◽  
Dongsheng Zhao
Keyword(s):  
Topological Space ◽  
Domain Theory ◽  
New Models ◽  
Different Types ◽  
Maximal Point ◽  
Order Structures ◽  
Maximal Points

In domain theory, by a poset model of a T1 topological space X we usually mean a poset P such that the subspace Max(P) of the Scott space of P consisting of all maximal points is homeomorphic to X. The poset models of T1 spaces have been extensively studied by many authors. In this paper we investigate another type of poset models: lower topology models. The lower topology ?(P) on a poset P is one of the fundamental intrinsic topologies on the poset, which is generated by the sets of the form P\?x, x ? P. A lower topology poset model (poset LT-model) of a topological space X is a poset P such that the space Max?(P) of maximal points of P equipped with the relative lower topology is homeomorphic to X. The studies of such new models reveal more links between general T1 spaces and order structures. The main results proved in this paper include (i) a T1 space is compact if and only if it has a bounded complete algebraic dcpo LT-model; (ii) a T1 space is second-countable if and only if it has an ?-algebraic poset LT-model; (iii) every T1 space has an algebraic dcpo LT-model; (iv) the category of all T1 space is equivalent to a category of bounded complete posets. We will also prove some new results on the lower topology of different types of posets.

scholarly journals p-topological andp-regular: dual notions in convergence theory

1999 ◽  
Vol 22 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Scott A. Wilde ◽  
D. C. Kent
Keyword(s):  
Natural Duality ◽  
Convergence Theory ◽  
Convergence Criteria ◽  
Convergence Space ◽  
Regular Convergence ◽  
Convergence Spaces ◽  
Simple Characterization ◽  
Topological Convergence

The natural duality between “topological” and “regular,” both considered as convergence space properties, extends naturally top-regular convergence spaces, resulting in the new concept of ap-topological convergence space. Taking advantage of this duality, the behavior ofp-topological andp-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criteria. Applications include the topological and regularity series of a convergence space.

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