scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

scholarly journals A note of almost continuous mappings and Baire spaces

1983 ◽  
Vol 6 (1) ◽  
pp. 197-199
Author(s):  
Jing Cheng Tong
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Continuous Mappings ◽  
Almost Continuous

We prove the following theorem: THEOREM. LetYbe a second countable, infiniteR0-space. If there are countably many open sets01,02,…,0n,…inYsuch that01⫋02⫋…⫋0n⫋…, then a topological spaceXis a Baire space if and only if every mappingf:X→Yis almost continuous on a dense subset ofX. It is an improvement of a theorem due to Lin and Lin [2].

Almost Continuity of Mappings

1968 ◽  
Vol 11 (3) ◽  
pp. 453-455 ◽  
Author(s):  
Shwu-Yeng T. Lin
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Direct Proof ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Natural Context ◽  
Set Of Points

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].

scholarly journals Another note on almost continous mappings and Baire spaces

1984 ◽  
Vol 7 (3) ◽  
pp. 619-620
Author(s):  
Jingcheng Tong
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Regular Open ◽  
Almost Continuous

The following result is proved:LetYbe a second countable, infinite topological space with an ascending chain of regular open sets. Then a topological spaceXis a Baire space if and only if every mappingf:X→Yis almost continuous on a dense subset ofX.It is another improvement of a theorem of Lin and Lin [2].

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.

\(\alpha_\beta\)-Connectedness as a characterization of connectedness

2018 ◽  
Vol 9 (2) ◽  
pp. 119-129 ◽  
Author(s):  
B. K. Tyagi ◽  
Manoj Bhardwaj ◽  
Sumit Singh
Keyword(s):  
Topological Space ◽  
Intermediate Value Theorem ◽  
New Class ◽  
Continuous Mappings ◽  
Intermediate Value ◽  
Connected Spaces

In this paper, a new class of \(\alpha_\beta\)-open sets in a topological space \(X\) is introduced which forms a topology on \(X\). The connectedness of this new topology on \(X\), called \(\alpha_\beta\)-connectedness, turns out to be equivalent to connectedness of \(X\) and hence also to \(\alpha\)-connectedness of \(X\). The \(\alpha_\beta\)-continuous and \(\alpha_\beta\)-irresolute mappings are defined and their relationship with other mappings such as continuous mappings and \(\alpha\)-continuous mappings are discussed. An intermediate value theorem is obtained. The hyperconnected spaces constitute a subclass of \(\alpha_\beta\)-connected spaces.

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.

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

scholarly journals Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.

A Note on Extending Locally Finite Collections

1976 ◽  
Vol 19 (1) ◽  
pp. 117-119
Author(s):  
H. L. Shapiro ◽  
F. A. Smith
Keyword(s):  
Topological Space ◽  
Set Cover ◽  
Topological Spaces ◽  
Locally Finite ◽  
Whole Space

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

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