On Homeomorphisms between Extension Spaces

1960 ◽  
Vol 12 ◽  
pp. 252-262 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
General Result ◽  
Hausdorff Space ◽  
Dimensional Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Minimal Extension ◽  
Completely Regular ◽  
General Terms ◽  
Extension Spaces ◽  
Homeomorphic Extension

In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).

scholarly journals Intuitionistic Fuzzy Topological Spaces Model

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Hind Fadhil Abbas
Keyword(s):  
Hausdorff Space ◽  
Basic Structure ◽  
Normal Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Scientific Phenomenon ◽  
Fuzzy Topological Spaces

The fusion of technology and science is a very complex and scientific phenomenon that still carries mysteries that need to be understood. To unravel these phenomena, mathematical models are beneficial to treat different systems with unpredictable system elements. Here, the generalized intuitionistic fuzzy ideal is studied with topological space. These concepts are useful to analyze new generalized intuitionistic models. The basic structure is studied here with various relations between the generalized intuitionistic fuzzy ideals and the generalized intuitionistic fuzzy topologies. This study includes intuitionistic fuzzy topological spaces (IFS); the fundamental definitions of intuitionistic fuzzy Hausdorff space; intuitionistic fuzzy regular space; intuitionistic fuzzy normal space; intuitionistic fuzzy continuity; operations on IFS, the compactness and separation axioms.

Hewitt Realcompactifications of Products

1970 ◽  
Vol 22 (3) ◽  
pp. 645-656 ◽  
Author(s):  
William G. McArthur
Keyword(s):  
General Result ◽  
Hausdorff Space ◽  
Ad Hoc ◽  
Completely Regular ◽  
Analogous Question ◽  
Open Question ◽  
Hewitt Realcompactification

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.

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.

On the Katětov and Stone-Čech Extensions

1959 ◽  
Vol 2 (1) ◽  
pp. 1-4 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Compact Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Approximation Theorem ◽  
General Topology ◽  
Natural Question ◽  
Completely Regular ◽  
Extension Spaces ◽  
Weierstrass Approximation Theorem

In general topology, one knows several standard extension spaces defined for one class of spaces or another and it is a natural question concerning any two such extensions which are defined for the same space whether they can ever be equal to each other. In the following, this problem will be considered for the Stone-Čech compactification βE of a completely regular non-compact Hausdorff space E[4] and Katětov's maximal Hausdorff extension κE of E[5]. It will be shown that βEκE always holds or, what amounts to the same, that κE can never be compact. As an application of this it will be proved that any completely regular Hausdorff space is dense in some non-compact space in which the Stone-Weierstrass approximation theorem holds.

scholarly journals 0-dimensional compactifications and Boolean rings

1968 ◽  
Vol 8 (4) ◽  
pp. 755-765 ◽  
Author(s):  
K. D. Magill ◽  
J. A. Glasenapp
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Hausdorff Space ◽  
Dimensional Space ◽  
Continuous Mapping ◽  
Dense Subspace ◽  
Clopen Subset ◽  
Partially Order ◽  
Completely Regular ◽  
Boolean Rings

A subset of a topological space which is both closed and open is referred to as a clopen subset. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. By a compactification αX of a completely regular Hausdorff space X, we mean any compact space which contains X as a dense subspace. Two compactifications αX and γX are regarded as being equivalent if there exists a homeomorphism from αX onto γX which keeps X pointwise fixed. We will not distinguish between equivalent compactifications. With this convention, we can partially order any family of compactifications of X by defining αX ≧ γX if there exists a continuous mapping from γX onto αX which leaves X pointwise fixed. This paper is concerned with the study of the partially ordered family [X] of all 0-dimensional compactifications of a 0-dimensional space X.

A characterization of completely regular spaces

2015 ◽  
Vol 26 (03) ◽  
pp. 1550032 ◽  
Author(s):  
Richard W. M. Alves ◽  
Victor H. L. Rocha ◽  
Josiney A. Souza
Keyword(s):  
Topological Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Admissible Family

This paper proves that uniform spaces and admissible spaces form the same class of topological spaces. This result characterizes a completely regular space as a topological space that admits an admissible family of open coverings. In addition, the admissible family of coverings provides an interesting methodology of studying aspects of uniformity and dynamics in completely regular spaces.

Developability and Some New Regularity Axioms

1981 ◽  
Vol 33 (3) ◽  
pp. 641-663 ◽  
Author(s):  
N. C. Heldermann
Keyword(s):  
Continuous Function ◽  
Closed Subset ◽  
Natural Extension ◽  
Unit Interval ◽  
Regularity Conditions ◽  
Regular Space ◽  
Topological Spaces ◽  
Completely Regular ◽  
Natural Way

In a recent publication H. Brandenburg [5] introduced D-completely regular topological spaces as a natural extension of completely regular (not necessarily T1) spaces: Whereas every closed subset A of a completely regular space X and every x ∈ X\A can be separated by a continuous function into a pseudometrizable space (namely into the unit interval), D-completely regular spaces admit such a separation into developable spaces. In analogy to the work of O. Frink [16], J. M. Aarts and J. de Groot [19] and others ([38], [46]), Brandenburg derived a base characterization of D-completely regular spaces, which gives rise in a natural way to two new regularity conditions, D-regularity and weak regularity.

Extensions of Topological Spaces

1964 ◽  
Vol 7 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Complex Number ◽  
Large Body ◽  
General Topology ◽  
General Question ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Wide Range ◽  
Extension Spaces ◽  
The One

The undertaking of constructing spaces which contain a given space as a subspace is by no means new: the extension of the complex number plane to the complex number sphere by the addition of the one point at infinity, the extension of the real line by adjoining the two infinities ∞ and -∞, and the construction of the space of real numbers from that of the rationals by means of Cauchy sequences or Dedekind cuts are 19th Century examples of this very thing. However, only the advent of general topology made it possible to raise the general question of space extensions. It appears that the first study of problems in this area was carried out by Alexandroff and Urysohn in the early twenties [l]. Another mile stone in the history of the subject was the 1929 paper by Tychonoff in which the product theorem for compact spaces is proved and used to identify the completely regular Hausdorff spaces as precisely those spaces which can be imbedded in a compact Hausdorff space [33]. During the same period, work on certain specific extension problems was done by Freudenthal [17] and Zippin [35]. However, the first large body of systematic theory, used for the investigation of a wide range of extension problems, was presented by Stone [31] in 1937. There, one also finds the remark that "one of the interesting and difficult problems of general topology is the study of all extensions of a given space", and it appears that Stone' s own work must have convinced many others of the truth of this observation, for since that time there has been a steady succession of papers in this field. But apart from that, the study of extension spaces clearly has a very particular attraction for some mathematicians.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

scholarly journals Two semigroups of continuous relations

1977 ◽  
Vol 23 (1) ◽  
pp. 46-58 ◽  
Author(s):  
A. R. Bednarek ◽  
Eugene M. Norris
Keyword(s):  
Large Class ◽  
Algebraic Structure ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Topological Semigroups ◽  
Stone Type

SynopsisIn this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

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