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.

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

Functional characterizations of p-spaces

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Ľubica Holá
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Open Subset ◽  
Continuous Extension ◽  
Regular Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Complete Space ◽  
Completely Regular ◽  
Locally Compact Space

AbstractWe show that a completely regular space Y is a p-space (a Čech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

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 Gelfand theorem implies Stone representation theorem of Boolean rings

1995 ◽  
Vol 18 (4) ◽  
pp. 701-704
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Boolean Rings ◽  
Stone Representation ◽  
Stone Theorem

Stone Theorem about representing a Boolean algebra in terms of open-closed subsets of a topological space is a consequence of the Gelfand Theorem about representing aB∗- algebra as the algebra of continuous functions on a compact Hausdorff space.

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 Multiplication operators on weighted spaces in the non-locally convex framework

1997 ◽  
Vol 20 (1) ◽  
pp. 75-79 ◽  
Author(s):  
L. A. Khan ◽  
A. B. Thaheem
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Weighted Space ◽  
Topological Algebra ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Completely Regular ◽  
General Topological Space ◽  
Necessary And Sufficient ◽  
Locally Convex

LetXbe a completely regular Hausdorff space,Ea topological vector space,Va Nachbin family of weights onX, andCV0(X,E)the weighted space of continuousE-valued functions onX. Letθ:X→Cbe a mapping,f∈CV0(X,E)and defineMθ(f)=θf(pointwise). In caseEis a topological algebra,ψ:X→Eis a mapping then defineMψ(f)=ψf(pointwise). The main purpose of this paper is to give necessary and sufficient conditions forMθandMψto be the multiplication operators onCV0(X,E)whereEis a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption thatEis locally convex.

Monadic binary relations and the monad systems at near-standard points

1987 ◽  
Vol 52 (3) ◽  
pp. 689-697
Author(s):  
Nader Vakil
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Hausdorff Space ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Completely Regular ◽  
General Topological Space ◽  
Image Position

AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

scholarly journals A characterization of pseudocompactness

1981 ◽  
Vol 4 (2) ◽  
pp. 407-409
Author(s):  
Prabduh Ram Misra ◽  
Vinodkumar
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Hausdorff Space ◽  
Completely Regular ◽  
Pseudocompact Space

It is proved here that a completely regular Hausdorff space X is pseudocompact if and only if for any continuous function f from X to a pseudocompact space (or a compact space) Y, f*ϕ is z-ultrafilter whenever ϕ is a z-ultrafilter on X.

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).

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