scholarly journals Delta – Open Sets And Delta – Continuous Functions

2021 ◽  
Vol 8 ◽  
pp. 1-23
Author(s):  
Raja Mohammad Latif
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Fundamental Properties ◽  
Closed Mappings

In 1968 Velicko [30] introduced the concepts of δ-closure and δ-interior operations. We introduce and study properties of δ-derived, δ-border, δ-frontier and δ-exterior of a set using the concept of δ-open sets. We also introduce some new classes of topological spaces in terms of the concept of δ-D- sets and investigate some of their fundamental properties. Moreover, we investigate and study some further properties of the well-known notions of δ-closure and δ-interior of a set in a topological space. We also introduce δ-R0 space and study its characteristics. We also introduce δ-R0 space and study its characteristics. We introduce δ-irresolute, δ-closed, pre-δ-open and pre -δ-closed mappings and investigate properties and characterizations of these new types of mappings and also explore further properties of the well-known notions of δ-continuous and δ-open mappings.

scholarly journals Constructing Banaschewski compactification without Dedekind completeness axiom

2004 ◽  
Vol 2004 (69) ◽  
pp. 3799-3816
Author(s):  
S. K. Acharyya ◽  
K. C. Chattopadhyay ◽  
Partha Pratim Ghosh
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Open Problems ◽  
Structure Space ◽  
Ordered Field ◽  
Completeness Axiom ◽  
Dedekind Completeness ◽  
Stone Theorem ◽  
Stone’S Theorem

The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero-dimensional Hausdorff topological space as a structure space of a ring of ordered field-valued continuous functions on the space, and thereby exhibit the independence of the construction from any completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone's theorem, the Banach-Stone theorem, and the Gelfand-Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all zero-dimensional but not strongly zero-dimensional Hausdorff topological spaces into three classes that are labeled by inequalities between three compactifications ofX, namely, the Stone-Čech compactificationβX, the Banaschewski compactificationβ0X, and the structure space𝔐X,Fof the lattice-ordered commutative ringℭ(X,F)of all continuous functions onXtaking values in the ordered fieldF, equipped with its order topology. Some open problems are also stated.

scholarly journals Fine Fuzzy sp Closed Sets in Fine Fuzzy Topological Spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 1306-1313
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Inverse Image ◽  
Continuous Functions ◽  
Topological Spaces ◽  
The Novel ◽  
Closed Set ◽  
Closed Sets ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The main view of this article is the extended version of the fine topological space to the novel kind of space say fine fuzzy topological space which is developed by the notion called collection of quasi coincident of fuzzy sets. In this connection, fine fuzzy closed sets are introduced and studied some features on it. Further, the relationship between fine fuzzy closed sets with certain types of fine fuzzy closed sets are investigated and their converses need not be true are elucidated with necessary examples. Fine fuzzy continuous function is defined as the inverse image of fine fuzzy closed set is fine fuzzy closed and its interrelations with other types of fine fuzzy continuous functions are obtained. The reverse implication need not be true is proven with examples. Finally, the applications of fine fuzzy continuous function are explained by using the composition.

scholarly journals On spaces defined by Pytkeev networks

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6115-6129 ◽  
Author(s):  
Xin Liu ◽  
Shou Lin
Keyword(s):  
Topological Space ◽  
Affirmative Answer ◽  
General Topology ◽  
Topological Spaces ◽  
Urysohn Space ◽  
Closed Mappings ◽  
The Relationship

The notions of networks and k-networks for topological spaces have played an important role in general topology. Pytkeev networks, strict Pytkeev networks and cn-networks for topological spaces are defined by T. Banakh, and S. Gabriyelyan and J. K?kol, respectively. In this paper, we discuss the relationship among certain Pytkeev networks, strict Pytkeev networks, cn-networks and k-networks in a topological space, and detect their operational properties. It is proved that every point-countable Pytkeev network for a topological space is a quasi-k-network, and every topological space with a point-countable cn-network is a meta-Lindel?f D-space, which give an affirmative answer to the following problem [25, 29]: Is every Fr?chet-Urysohn space with a pointcountable cs'-network a meta-Lindel?f space? Some mapping theorems on the spaces with certain Pytkeev networks are established and it is showed that (strict) Pytkeev networks are preserved by closed mappings and finite-to-one pseudo-open mappings, and cn-networks are preserved by pseudo-open mappings, in particular, spaces with a point-countable Pytkeev network are preserved by closed mappings.

scholarly journals Some remarks on fuzzy infi topological spaces

2021 ◽  
Vol 40 (2) ◽  
pp. 399-415
Author(s):  
Birojit Das ◽  
Baby Bhattacharya ◽  
Apu Kumar Saha
Keyword(s):  
Topological Space ◽  
Applied Mathematics ◽  
Topological Spaces ◽  
Italian Journal ◽  
Study Product ◽  
Fuzzy Topological Spaces ◽  
Closed Mappings

Induced fuzzy infi topological space is already introduced by Saha and Bhattacharya [Saha A.K., Bhattacharya D. 2015, Normal Induced Fuzzy Topological Spaces, Italian Journal of Pure and Applied Mathematics, 34, 45-56]. In this paper for the said space, we further analyse some properties viz. fuzzy I-continuity, fuzzy infi open mappings and fuzzy infi closed mappings etc. Also we study product fuzzy infi topological space and establish some results concerned with it.

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

Uniformities on a Product

1972 ◽  
Vol 24 (3) ◽  
pp. 379-389 ◽  
Author(s):  
Anthony W. Hager
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
The Real ◽  
Completely Regular

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

scholarly journals Continuous functions between sets with operations

2020 ◽  
Vol 24 (2) ◽  
pp. 225-239
Author(s):  
Fumie Nakaoka ◽  
Nobuyuki Oda
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Compact Spaces ◽  
Fundamental Properties ◽  
Connected Spaces

A set with an operation is a generalization of a topological space. Two types of continuous functions are dened between sets with operations. They are characterized making use of two types of closures and interiors. Homeomorphisms between sets with operations are also characterized. Variants of subspaces, connected spaces and compact spaces are introduced in a set with an operation and some fundamental properties of them are proved.

scholarly journals Beta- Star- Continuity and Beta- Star- Contra- Continuity

2021 ◽  
Vol 7 ◽  
pp. 43-66
Author(s):  
Raja Mohammad Latif
Keyword(s):  
Continuous Functions ◽  
General Topology ◽  
Topological Spaces ◽  
New Class ◽  
Fundamental Properties ◽  
Open Set ◽  
Class Of Functions

In 2014 Mubarki, Al-Rshudi, and Al- Juhani introduced and studied the notion of a set in general topology called β*-open set and investigated its fundamental properties and studied the relationships between β*-open set and other topological sets including β*-continuity in topological spaces. We introduce and investigate several properties and characterizations of a new class of functions between topological spaces called β*- open, β*- closed, β*- continuous and β*- irresolute functions in topological spaces. We also introduce slightly β*- continuous, totally β*- continuous and almost β*- continuous functions between topological spaces and establish several characterizations of these new forms of functions. Furthermore, we also introduce and investigate certain ramifications of contra continuous and allied functions, namely, contra β*- continuous, and almost contra β*-continuous functions along with their several properties, characterizations and natural relationships. Moreover, we introduce new types of closed graphs by using β*- open sets and investigate its properties and characterizations in topological spaces.

scholarly journals Co-Compact Separation Axioms and Slight Co-Continuity

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1614
Author(s):  
Samer Al Ghour ◽  
Enas Moghrabi
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Closed Subspace ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Separation Axioms ◽  
Preservation Theorems

Via co-compact open sets we introduce co-T2 as a new topological property. We show that this class of topological spaces strictly contains the class of Hausdorff topological spaces. Using compact sets, we characterize co-T2 which forms a symmetry. We show that co-T2 propoerty is preserved by continuous closed injective functions. We show that a closed subspace of a co-T2 topological space is co-T2. We introduce co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. We obtain several characterizations, implications, and examples regarding co-regularity, s-regularity and co-normality. Moreover, we give several preservation theorems under slightly coc-continuous functions.

I-Subsemigroups and α-monomorphisms

1973 ◽  
Vol 16 (2) ◽  
pp. 146-166 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Natural Way

To each idempotent v of a semigroup T, there is associated, in a natural way, a subsemigroup Tv of T. The subsemigroup Tv is simply the collection of all elements of T for which v acts as a two-sided identity. We refer to such a subsemigroup as an I-subsemigroup of T. We first establish some elementary properties of these subsemigroups with no restrictions on the semigroup in which they are contained. Then we turn our attention to the semigroup of all continuous selfmaps of a topological space. The I-subsemigroups of these semigroups are investigated in some detail and so are the a-monomorphisms [3, p. 518] from one such semigroup into another. Among other things, a relationship is established between I-subsemigroups and α-monomorphisms. An analogous theory exists for semigroups of closed selfmaps on topological spaces. A number of results are listed for these semigroups with the proofs often deleted since, in many cases, the situation is much the same as for semigroups of continuous functions.

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