scholarly journals A Characterization of the discontinuity point set of strongly separately continuous functions on products

2016 ◽  
Vol 66 (6) ◽  
Author(s):  
Olena Karlova ◽  
Volodymyr Mykhaylyuk
Keyword(s):  
Continuous Mapping ◽  
Discontinuity Point ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Finite Dimensional ◽  
Continuous Mappings ◽  
Countable Product ◽  
Topology Of Pointwise Convergence ◽  
Point Set ◽  
Necessary And Sufficient

AbstractWe study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a strongly separately continuous mapping to be continuous on a product of an arbitrary family of topological spaces. Moreover, we characterize the discontinuity point set of strongly separately continuous function defined on a subset of countable product of finite-dimensional normed spaces.

scholarly journals ‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

2019 ◽  
pp. 13
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎. 

scholarly journals Noncommutative Lattices and the Algebras of Their Continuous Functions

1998 ◽  
Vol 10 (04) ◽  
pp. 439-466 ◽  
Author(s):  
Elisa Ercolessi ◽  
Giovanni Landi ◽  
Paulo Teotonio-Sobrinho
Keyword(s):  
Quantum Physics ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Hausdorff Topology ◽  
Topological Information ◽  
C Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
Af Algebras ◽  
The Continuum

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

The pseudocompactness of [0.1] is equivalent to the uniform continuity theorem

2007 ◽  
Vol 72 (4) ◽  
pp. 1379-1384 ◽  
Author(s):  
Douglas Bridges ◽  
Hannes Diener
Keyword(s):  
Metric Space ◽  
Continuous Mapping ◽  
Uniform Continuity ◽  
Compact Metric Space ◽  
Continuity Theorem ◽  
Continuous Mappings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

AbstractWe prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

scholarly journals SOFT ALMOST β-CONTINUITY IN SOFT TOPOLOGICAL SPACES

2020 ◽  
Vol 8 (2) ◽  
pp. 06-14
Author(s):  
Alpa Singh Rajput ◽  
S. S. Thakur ◽  
Om Prakash Dubey
Keyword(s):  
Image Segmentation ◽  
Information Systems ◽  
Mathematical Models ◽  
General Framework ◽  
Continuous Mapping ◽  
Topological Spaces ◽  
Major Area ◽  
Soft Topology ◽  
Continuous Mappings

Purpose: In the present paper the concept of soft almost β-continuous mappings and soft almost β-open mappings in soft topological spaces have been introduced and studied. Methodology: This notion is weaker than both soft almost pre-continuous mappings, soft almost semi-continuous mapping. The diagrams of implication among these soft classes of soft mappings have been established. Main Findings: We extend the concept of almost β-continuous mappings and almost β-open mappings in soft topology. Implications: Mapping is an important and major area of topology and it can give many relationships between other scientific areas and mathematical models. This notion captures the idea of hanging-togetherness of image elements in an object by assigning strength of connectedness to every possible path between every possible pair of image elements. It is an important tool for the designing of algorithms for image segmentation. The novelty of Study: Hope that the concepts and results established in this paper will help the researcher to enhance and promote the further study on soft topology to carry out a general framework for the development of information systems.

scholarly journals On Strongly b − θ -Continuous Mappings in Fuzzifying Topology

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Ting Yang ◽  
Ahmed Mostafa Khalil
Keyword(s):  
Fuzzy Logic ◽  
Continuous Mapping ◽  
Topological Spaces ◽  
Closed Set ◽  
Open Set ◽  
Neighborhood System ◽  
Continuous Mappings ◽  
Derived Set ◽  
New Type

In this article, we will define the new notions (e.g., b − θ -neighborhood system of point, b − θ -closure (interior) of a set, and b − θ -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b − θ -closed set is fuzzifying γ -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b − θ -derived set and fuzzifying b − θ -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b − θ -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b − θ -continuous mapping, fuzzifying strongly b − θ -irresolute mapping, and fuzzifying weakly b − θ -irresolute mapping along with different conditions for their existence are obtained.

scholarly journals The discontinuity point sets of quasi-continuous functions

2007 ◽  
Vol 75 (3) ◽  
pp. 373-379 ◽  
Author(s):  
Oleksandr V. Maslyuchenko
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Discontinuity Point ◽  
Continuous Functions ◽  
Countable Union ◽  
Point Sets ◽  
Point Set

It is proved that a subsetEof a hereditarily normal topological spaceXis a discontinuity point set of some quasi-continuous functionf:X→ ℝ if and only ifEis a countable union of setsEn=Ān⋂Bdash abovenwhereĀn⋂Bn=An⋂Bdash aboven= φ

scholarly journals Compactness in intuitionistic fuzzy topological spaces

2005 ◽  
Vol 2005 (1) ◽  
pp. 19-32 ◽  
Author(s):  
A. A. Ramadan ◽  
S. E. Abbas ◽  
A. A. Abd El-Latif
Keyword(s):  
Topological Space ◽  
Continuous Mapping ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Continuous Mappings ◽  
Weakly Continuous ◽  
Fuzzy Topological Space ◽  
Definition Of ◽  
Fuzzy Topological Spaces ◽  
Almost Continuous

We introduce fuzzy almost continuous mapping, fuzzy weakly continuous mapping, fuzzy compactness, fuzzy almost compactness, and fuzzy near compactness in intuitionistic fuzzy topological space in view of the definition of Šostak, and study some of their properties. Also, we investigate the behavior of fuzzy compactness under several types of fuzzy continuous mappings.

scholarly journals Fuzzy Almost Generalized E-Continuous Mappings in Smooth Topological Spaces

2019 ◽  
Vol 8 (10S) ◽  
pp. 138-141 ◽  
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuous Mapping ◽  
Regular Space ◽  
Topological Spaces ◽  
Continuous Mappings ◽  
Fuzzy Topological Space ◽  
Index Terms ◽  
Almost Continuous

We introduce and study several interesting properties of fuzzy almost generalized e -continuous mappings in smooth topological spaces with counter examples. We also introduce fuzzy r - 1 2 fT e -space, r -fuzzy ge -space, r -fuzzy regular ge -space and r -fuzzy generalized e -compact space. It is seen that a fuzzy almost generalized e -continuous mapping, between a fuzzy r - 1 2 fT e -space and a fuzzy topological space, becomes fuzzy almost continuous mapping. Index Terms: fage -continuous, r - fge -space, r - fge -regular space, r - 1 2 fT e -space.

scholarly journals On monotonous separately continuous functions

2019 ◽  
Vol 20 (1) ◽  
pp. 75
Author(s):  
Yaroslav I. Grushka
Keyword(s):  
Topological Space ◽  
Cartesian Product ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Order Topology ◽  
Topological Spaces ◽  
Ordered Sets ◽  
Tychonoff Topology ◽  
And Function

<p>Let T = (<strong>T</strong>, ≤) and T<sub>1</sub>= (<strong>T</strong><sub>1</sub> , ≤<sub>1</sub>) be linearly ordered sets and X be a topological space.  The main result of the paper is the following: If function ƒ(t,x) : <strong>T</strong> × X → <strong>T</strong><sub>1 </sub>is continuous in each  variable (“t” and  “x”)  separately  and  function ƒ<sub>x</sub>(t)  = ƒ(t,x) is  monotonous  on <strong>T</strong> for  every x ∈ X,  then ƒ is  continuous  mapping  from<strong> T</strong> × X to <strong>T</strong><sub>1</sub>,  where <strong>T</strong> and <strong>T</strong><sub>1</sub> are  considered  as  topological  spaces  under  the order topology and <strong>T</strong> × X is considered as topological space under the Tychonoff topology on the Cartesian  product of topological spaces <strong>T</strong> and X.</p>

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

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