scholarly journals A NOTE ON RELATIVE PSEUDOCOMPACTNESS IN THE CATEGORY OF FRAMES

2012 ◽  
Vol 87 (1) ◽  
pp. 120-130 ◽  
Author(s):  
THEMBA DUBE
Keyword(s):  
Tychonoff Space ◽  
Completely Regular ◽  
Tychonoff Spaces

AbstractA subspace S of Tychonoff space X is relatively pseudocompact in X if every f∈C(X) is bounded on S. As is well known, this property is characterisable in terms of the functor υ which reflects Tychonoff spaces onto the realcompact ones. A device which exists in the category CRegFrm of completely regular frames which has no counterpart in Tych is the functor which coreflects completely regular frames onto the Lindelöf ones. In this paper we use this functor to characterise relative pseudocompactness.

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.

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 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.

scholarly journals Strong 0-dimensionality in Pointfree Topology

10.29007/5dmr ◽  
2018 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Tychonoff Space ◽  
Completely Regular ◽  
Regular Frame ◽  
Completely Regular Frame

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Cech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen immediately that a Tychonoff space is strongly 0-dimensional iff the frame of its open sets is strongly 0-dimensional in the present sense. This talk will provide an account of various aspects of this notion.

scholarly journals Separability axioms and metric-like functions

1976 ◽  
Vol 15 (2) ◽  
pp. 265-271
Author(s):  
Howard Anton
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Completely Regular ◽  
Closed Sets ◽  
Tychonoff Spaces

We call a topological space completely regular if points and closed sets can be separated by continuous real-valued functions, and we call a topological space Tychonoff or T3α if it is T1 and completely regular. It is well known that the Tychonoff spaces are precisely the gauge spaces, that is the topological spaces whose topologies are induced by separating families of pseudometrics. V.G. Boltjanskiĭ has given analogous characterizations for T0, T1, T2, regular, and regular-T1 spaces in terms of families of “metric-like” functions. The purpose of this note is to fill in a case missing in Boltjanskiĭ's paper, the completely regular spaces, and to relate Boltjanskiĭ's work to results on quasi-uniformization by William J. Pervin and results about semi-gauge spaces by J.V. Michalowicz.

scholarly journals Characterized Fuzzy R2.5 and Characterized Fuzzy T3.5 Spaces

2017 ◽  
Vol 13 (1) ◽  
pp. 7048-7073
Author(s):  
Ahmed Saeed Abd-Allah
Keyword(s):  
Topological Space ◽  
Tychonoff Space ◽  
Proximity Space ◽  
Completely Regular ◽  
Fuzzy Function ◽  
Continuous Mappings ◽  
Fuzzy Space ◽  
Fuzzy Topological Space ◽  
The Given ◽  
Proximity Spaces

This paper, deals with, introduce and study the notions of haracterized fuzzy R2.5 spaces and of characterized fuzzy T3.5 spaces by using the notion of fuzzy function family presented in [21] and the notion of φ1,2ψ1,2-fuzzy continuous mappings presented in [5] as a generalization of all the weaker and stronger forms of the fuzzy completely regular spaces introduced in [11,24,26,29]. We denote by characterized fuzzy T3.5 space or characterized fuzzy Tychonoff space to the characterized fuzzy space which is characterized fuzzy T1 and characterized fuzzy R2.5 space in this sense. The relations between the characterized fuzzy T3.5 spaces, the characterized fuzzy T4 spaces and the characterized fuzzy T3 spaces are introduced. When the given fuzzy topological space is normal, then the related characterized fuzzy space is finer than the associated characterized fuzzy proximity space which is presented in [1]. Moreover, the associated characterized fuzzy proximity spaces and the characterized fuzzy T4 spaces are identical with help of the complementarilysymmetric fuzzy topogenous structure, that is, identified with the fuzzy proximity δ. More generally, the fuzzy function family of all φ1,2ψ1,2-fuzzy continuous mappings are applied to show that the characterized fuzzy R2.5 spaces and the associated characterized fuzzy proximity spaces are identical.

scholarly journals Maximal Tychonoff spaces and normal isolator covers

2016 ◽  
Vol 99 (113) ◽  
pp. 217-225
Author(s):  
C.K. Basu ◽  
S.S. Mandal
Keyword(s):  
Alternative Proof ◽  
Tychonoff Space ◽  
Extremally Disconnected ◽  
Tychonoff Spaces

We introduce a new kind of cover called a normal isolator cover to characterize maximal Tychonoff spaces. Such a study is used to provide an alternative proof of an interesting result of Feng and Garcia-Ferreira in 1999 that every maximal Tychonoff space is extremally disconnected. Maximal tychonoffness of subspaces is also discussed.

scholarly journals General rings of functions

1975 ◽  
Vol 20 (3) ◽  
pp. 359-365 ◽  
Author(s):  
A. Sultan
Keyword(s):  
General Setting ◽  
Continuous Functions ◽  
Tychonoff Space ◽  
Zero Sets ◽  
Rings Of Continuous Functions ◽  
Hewitt Realcompactification ◽  
Tychonoff Spaces

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

scholarly journals On maps between Stone-Cech compactifications induced by lattice homomorphisms

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2465-2474 ◽  
Author(s):  
Themba Dube
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lattice Homomorphism ◽  
Completely Regular ◽  
Zero Sets ◽  
Continuous Map ◽  
Necessary And Sufficient ◽  
Lattice Homomorphisms ◽  
Tychonoff Spaces

Broverman has shown that if X and Y are Tychonoff spaces and t:Z(Y)?Z(X) is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map ?: ?X ? ?Y induced by t. In this note we expound this idea and supplement Broverman?s results by first showing that this phenomenon holds in the category of completely regular frames. Among results we obtain, which were not considered by Broverman, are necessary and sufficient conditions (in terms of properties of the map t) for the induced map ? to be (i) the inclusion of a subspace, (ii) surjective, and (iii) irreducible. We show that if X and Y are pseudocompact then t pulls back z-ultrafilters to z-ultrafilters if and only if cl?X t(Z) = ?? [cl?YZ] for every Z ? Z(Y) if and only if t is ?-homomorphism.

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