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 A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

2019 ◽  
Vol 69 (5) ◽  
pp. 979-988
Author(s):  
Jissy Nsonde Nsayi
Keyword(s):  
Closed Subset ◽  
Natural Extension ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Regular Closed

Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

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 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 Distinguished Property in Tensor Products and Weak* Dual Spaces

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 151
Author(s):  
Salvador López-Alfonso ◽  
Manuel López-Pellicer ◽  
Santiago Moll-López
Keyword(s):  
Convex Space ◽  
Tensor Products ◽  
Tychonoff Space ◽  
Bounded Set ◽  
Injective Tensor Product ◽  
Tensor Product Space ◽  
Pointwise Topology ◽  
Local Convex ◽  
Strong Dual

A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided.

scholarly journals Characterization of pseudocompactness by the topology of uniform convergence on function spaces

1978 ◽  
Vol 26 (2) ◽  
pp. 251-256 ◽  
Author(s):  
R. A. McCoy
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Metrizable Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Tychonoff Space ◽  
Necessary And Sufficient

AbstractIt is shown that a Tychonoff space X is pseudocompact if and only if for every metrizable space Y, all uniformities on Y induce the same topology on the space of continuous functions from X into Y. Also for certain pairs of spaces X and Y, a necessary and sufficient condition is established in order that all uniformities on Y induce the same topology on the space of continuous functions from X into Y.

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.

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