The Maximal Extension of a Zero-dimensional Product Space

1983 ◽  
Vol 26 (2) ◽  
pp. 192-201
Author(s):  
Haruto Ohta
Keyword(s):  
Product Space ◽  
Topological Property ◽  
Continuous Extension ◽  
Dense Subspace ◽  
Hausdorff Spaces ◽  
Maximal Extension ◽  
Tychonoff Space ◽  
Continuous Map ◽  
Countable Compactness ◽  
Tychonoff Spaces

AbstractIt is known that if a topological property of Tychonoff spaces is closed-hereditary, productive and possessed by all compact Hausdorff spaces, then each (0-dimensional) Tychonoff space X is a dense subspace of a (0-dimensional) Tychonoff space with such that each continuous map from X to a (0-dimensional) Tychonoff space with admits a continuous extension over . In response to Broverman's question [Canad. Math. Bull. 19 (1), (1976), 13–19], we prove that if for every two 0-dimensional Tychonoff spaces X and Y, if and only if , then is contained in countable compactness.

scholarly journals Bounded resolutions for spaces $$C_{p}(X)$$ and a characterization in terms of X

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. C. Ferrando ◽  
J. Ka̧kol ◽  
W. Śliwa
Keyword(s):  
Tychonoff Space ◽  
Cosmic Space ◽  
Tychonoff Spaces

AbstractAn internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that the space $$C_{p}(X)$$ C p ( X ) of continuous real-valued functions on a Tychonoff space X is K-analytic framed in $$\mathbb {R}^{X}$$ R X if and only if X admits a nice framing. This applies to show that a metrizable (or cosmic) space X is $$\sigma $$ σ -compact if and only if X has a nice framing. We analyse a few concepts which are useful while studying nice framings. For example, a class of Tychonoff spaces X containing strictly Lindelöf Čech-complete spaces is introduced for which a variant of Arkhangel’skiĭ-Calbrix theorem for $$\sigma $$ σ -boundedness of X is shown.

scholarly journals Compactifications of totally bounded quasi-uniform spaces

1986 ◽  
Vol 28 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P. Fletcher ◽  
W. F. Lindgren
Keyword(s):  
Hausdorff Space ◽  
Uniform Space ◽  
Continuous Extension ◽  
Sufficient Condition ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Totally Bounded ◽  
Continuous Map ◽  
Total Boundedness ◽  
Uniformly Continuous

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

scholarly journals On compactification of mappings

1974 ◽  
Vol 19 (2) ◽  
pp. 105-108
Author(s):  
P. A. Firby
Keyword(s):  
Continuous Function ◽  
Dense Subspace ◽  
Compact Extension ◽  
Tychonoff Spaces ◽  
Function Mapping

If X and Y are Tychonoff spaces then the continuous function f mapping X onto Y is said to be compact (perfect, or proper) if it is closed and point inverses are compact. If h is a continuous function mapping X onto Y then by a compactification of h we mean a pair (X*, h*) where X* is Tychonoff and contains X as a dense subspace, and where h*: X*→Y is a compact extension of h. The idea of a mapping compactification first appeared in (7). In (1) it was shown that any compactification of X determines a compactification of h, and that any compactification of h can be determined in this way. This idea was then developed in (2) and (3).

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.

scholarly journals Absolutes of almost realcompactifications

1986 ◽  
Vol 41 (2) ◽  
pp. 251-267
Author(s):  
Mohan L. Tikoo
Keyword(s):  
Hausdorff Spaces ◽  
Continuous Map ◽  
Natural Way

AbstractGiven Hausdorff spaces X and Y and a perfect irreducible and θ-continuous map f from X onto Y, technique that carries open (ultra) filters on X to open (ultra) filters on Y back and forth in a natural way is introduced. It is proved that if f is a perfect irreducible and θ-continuous map from X onto Y, then X is almost realcompact if and only if Y is almost realcompact. Several commutativity relations between the ‘absolutes of almost realcompactifications’ and the ‘almost realcompactifications of absólutes’ of a space X are discussed.

scholarly journals p-topological Cauchy completions

1999 ◽  
Vol 22 (3) ◽  
pp. 497-509
Author(s):  
J. Wig ◽  
D. C. Kent
Keyword(s):  
Equivalence Class ◽  
Topological Property ◽  
Convergence Space ◽  
Extension Theorems ◽  
General Properties ◽  
Continuous Map ◽  
Cauchy Space ◽  
Topological Completion ◽  
Cauchy Spaces ◽  
Natural Way

The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular,p-regular, and topological Cauchy completions, we hereby initiate a study ofp-topological Cauchy completions. Ap-topological Cauchy space has ap-topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing ap-topological completion, it is shown that a certain class of Reed completions preserve thep-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsestp-topological completions. However, not allp-topological completions are Reed completions. Several extension theorems forp-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowingp-topological andp′-topological completions, respectively, can always be extended to aθ-continuous map between anyp-topological completion of the first space and anyp′-topological completion of the second.

Characterizations of functionally compact spaces

1980 ◽  
Vol 29 (1) ◽  
pp. 71-79
Author(s):  
L. L. Herrington
Keyword(s):  
Topological Property ◽  
Compact Spaces ◽  
Countable Compactness

AbstractThis paper gives characterizations of functionally compact spaces in terms of filterbases and nets. Also, a topological property that is weaker than countable compactness but stronger than first countable minimality is investigated.

Reflexive objects in topological categories

1996 ◽  
Vol 6 (4) ◽  
pp. 375-386
Author(s):  
Michael D. Rice
Keyword(s):  
Geometric Structure ◽  
Unit Interval ◽  
Continuous Image ◽  
Topological Category ◽  
Hausdorff Spaces ◽  
Tychonoff Space ◽  
Countably Compact ◽  
Discrete Subspace ◽  
Cartesian Closed

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

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 On the maximalG-compactification of products of twoG-spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Natella Antonyan
Keyword(s):  
Topological Group ◽  
Tychonoff Space ◽  
Tychonoff Spaces

LetGbe any Hausdorff topological group and letβGXdenote the maximalG-compactification of aG-Tychonoff spaceX. We prove that ifXandYare twoG-Tychonoff spaces such that the productX×Yis pseudocompact, thenβG(X×Y)=βGX×βGX.

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