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.

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.

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.

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 Graph theoretic representation of rings of continuous functions

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3417-3428
Author(s):  
Bedanta Bose ◽  
Angsuman Das
Keyword(s):  
Continuous Functions ◽  
Topological Spaces ◽  
Graph Structure ◽  
Zero Set ◽  
Tychonoff Space ◽  
Graph Theoretic ◽  
Maximal Ideals ◽  
Set Intersection ◽  
Algebraic Properties ◽  
Relationship Of

In this paper, we introduce a graph structure, called zero-set intersection graph ?(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the inter-relationship of cliques of ?(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of ?(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.

scholarly journals On regular ideals in reduced rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3715-3726 ◽  
Author(s):  
A.R. Aliabady ◽  
R. Mohamadian ◽  
S. Nazari
Keyword(s):  
Closed Subset ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Closed Set ◽  
Von Neumann ◽  
Arbitrary Ring ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Isolated Points ◽  
Maximal Regular Ideal

Let R be a commutative ring with identity and X be a Tychonoff space. An ideal I of R is Von Neumann regular (briefly, regular) if for every a ? I, there exists b ? R such that a = a2b. In the present paper, we obtain the general form of a regular ideal in C(X) which is OA, for some closed subset A of ?X, for which Ac?X ? (P(X))?, where P(X) is the set of all P-points of X. We show that the ideals and subrings such as CK(X), C?(X), C?(X), SocmC(X) and M?X\X are regular if and only if they are equal to the socle of C(X). We carry further the study of the maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them OPD-spaces) the maximal regular ideal is OX\I(X), where I(X) is the set of isolated points of X. Also, for this class, the socle of C(X) is the maximal regular ideal if and only if I(X) contains no infinite closed set. We also show that C(X) contains an ideal which is both essential and regular if and only if (P(X))? is dense in X. Finally it is shown that, for semiprimitive rings pure ideals are of the form OA which A is a closed subset of Max(R), also a P-point of X = Max(R) is introduced and it is shown that the maximal regular ideal of an arbitrary ring R is OX\P(X), which P(X) is the set of P-points of X = Max(R).

scholarly journals On product of spaces of quasicomponents

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6307-6311
Author(s):  
Gjorgji Markoski ◽  
Abdulla Buklla
Keyword(s):  
Product Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Topological Properties ◽  
Locally Compact

We use a characterization of quasicomponents by continuous functions to obtain the well known theorem which states that product of quasicomponents Qx,Qy of topological spaces X,Y, respectively, gives quasicomponent in the product space X x Y. If spaces X,Y are locally-compact, paracompact and Haussdorf, then we prove that the space of quasicomponents of the product Q(XxY) is homeomorphic with the product space Q(X) x Q(Y), so these two spaces have the same topological properties.

scholarly journals Pseudo-idempotents in semigroups of functions

1974 ◽  
Vol 18 (2) ◽  
pp. 182-187
Author(s):  
Frank A. Cezus
Keyword(s):  
Main Part ◽  
Continuous Functions ◽  
Discrete Space ◽  
Topological Spaces ◽  
Binary Relations ◽  
Generalize Theorem ◽  
Paper Form

The aim of this paper is to generalize Theorem 2.10 (i) of [2]. As stated in [2] this theorem deals with the semigroup of all selfmaps on a discrete space and provides a characterization of H-classes which contain an idempotent. We will generalize this theorem to the case of other semigroups of functions on a discrete space, some semigroups of continuous functions on non-discrete topological spaces, and one semigroup of binary relations. The results in this paper form the main part of chapter 3 of [1]. Some results will be quoted from [1] without proof; the required proofs can easily be supplied by the reader.

scholarly journals F-Rings of Continuous Functions I

1959 ◽  
Vol 11 ◽  
pp. 80-86 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
The Other ◽  
Topological Spaces ◽  
Closed Sets ◽  
Real Complex

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

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