scholarly journals A note on some applications ofα-open sets

2003 ◽  
Vol 2003 (2) ◽  
pp. 125-130 ◽  
Author(s):  
Miguel Caldas
Keyword(s):  
Continuous Functions ◽  
Topological Properties

The object of this note is to introduce and study topological properties ofα-derived,α-border,α-frontier, andα-exterior of a set using the concept ofα-open sets. Moreover, we study some further properties of the well-known notions ofα-closure andα-interior. We also obtain a new decomposition ofα-continuous functions.

scholarly journals On generalizations of C*-embedding for wallman rings

1978 ◽  
Vol 25 (2) ◽  
pp. 215-229 ◽  
Author(s):  
H. L. Bentley ◽  
B. J. Taylor
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Properties ◽  
Zero Sets ◽  
Algebraic Properties

AbstractBiles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on X in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z(A). Here we introduce two different generalizations of the concept of “a C*-embedded subset” and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions

1980 ◽  
Vol 32 (3) ◽  
pp. 657-685 ◽  
Author(s):  
F. Dashiell ◽  
A. Hager ◽  
M. Henriksen
Keyword(s):  
Continuous Functions ◽  
Topological Properties ◽  
Order Convergence ◽  
Sequential Order ◽  
Compact Spaces ◽  
Lattice Group ◽  
Vector Lattices ◽  
Topological Condition ◽  
Bounded Continuous Functions ◽  
Tychonoff Spaces

This paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified, for compact spaces X, in [6]. This condition is that every dense cozero set S in X should be C*-embedded in X (that is, all bounded continuous functions on S extend to X). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the F-spaces of [12]).In Section 1, the notion of a completion with respect to sequential order convergence is first described in the setting of a commutative lattice group G.

scholarly journals Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Manuel Felipe Cerpa-Torres ◽  
Michael A. Rincón-Villamizar
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Geometrical Parameter ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Continuous Image ◽  
Topological Properties ◽  
Locally Compact ◽  
Regular Subspace ◽  
Locally Compact Hausdorff Space

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

scholarly journals On the Fuzzy Number Space with the Level Convergence Topology

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. J. Font ◽  
A. Miralles ◽  
M. Sanchis
Keyword(s):  
Topological Space ◽  
Fuzzy Number ◽  
Continuous Functions ◽  
Topological Properties ◽  
Compact Sets ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Convergence Topology

We characterize compact sets of𝔼1endowed with the level convergence topologyτℓ. We also describe the completion(𝔼1̂,𝒰̂)of𝔼1with respect to its natural uniformity, that is, the pointwise uniformity𝒰, and show other topological properties of𝔼1̂, as separability. We apply these results to give an Arzela-Ascoli theorem for the space of(𝔼1,τℓ)-valued continuous functions on a locally compact topological space equipped with the compact-open topology.

SIMPLICITY OF CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS

2019 ◽  
Vol 108 (2) ◽  
pp. 202-225
Author(s):  
ALEXANDRE BARAVIERA ◽  
WAGNER CORTES ◽  
MARLON SOARES
Keyword(s):  
Hausdorff Space ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Products ◽  
Topological Properties ◽  
Partial Action ◽  
Necessary And Sufficient ◽  
Skew Group Ring

In this article, we consider a twisted partial action $\unicode[STIX]{x1D6FC}$ of a group $G$ on an associative ring $R$ and its associated partial crossed product $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$. We provide necessary and sufficient conditions for the commutativity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ when the twisted partial action $\unicode[STIX]{x1D6FC}$ is unital. Moreover, we study necessary and sufficient conditions for the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ in the following cases: (i) $G$ is abelian; (ii) $R$ is maximal commutative in $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$; (iii) $C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$ is simple; (iv) $G$ is hypercentral. When $R=C_{0}(X)$ is the algebra of continuous functions defined on a locally compact and Hausdorff space $X$, with complex values that vanish at infinity, and $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ is the associated partial skew group ring of a partial action $\unicode[STIX]{x1D6FC}$ of a topological group $G$ on $C_{0}(X)$, we study the simplicity of $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ by using topological properties of $X$ and the results about the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$.

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 On quasi I-openness and quasi I-continuity

2000 ◽  
Vol 31 (2) ◽  
pp. 101-108
Author(s):  
M. E. Abd El-Monsef ◽  
R. A. Mahmoud ◽  
A. A. Nasef
Keyword(s):  
Continuous Functions ◽  
Finite Additivity ◽  
Topological Properties ◽  
Closed Sets ◽  
New Concepts

A space $ (X,\tau,I)$ consisting of a nonempty set $ X$ with a topology $ \tau$ and an ideal $ I$ of subsets of $ X$ which has heredity and finite additivity properties. In this paper the quasi $ I$-open and quasi $ I$-closed sets are presented. Utilizing these new concepts the class of quasi $ I$-continuous functions have been obtained. Both of quasi $ I$-openness and quasi $ I$-continuity is considered as a generalization of those $ I$-openness and $ I$-continuity. However, numerous topological properties of these new notions have been discussed as well as many of their known results have been improved.

A Definition of Separation Axiom

1974 ◽  
Vol 17 (4) ◽  
pp. 485-491
Author(s):  
T. Cramer
Keyword(s):  
Continuous Functions ◽  
General Definition ◽  
Topological Properties ◽  
Separation Axiom ◽  
Separation Axioms ◽  
Definition Of

Several separation axioms, defined in terms of continuous functions, were examined by van Est and Freudenthal [3], in 1951. Since that time, a number of new topological properties which were called separation axioms were defined by Aull and Thron [1], and later by Robinson and Wu [2], This paper gives a general definition of separation axiom, defined in terms of continuous functions, and shows that the standard separation axioms, and all but one of these new topological properties, fit this definition.

A Topological Approach in the Extended Fraenkel-Mostowski Model of Set Theory

2014 ◽  
Vol 60 (2) ◽  
pp. 261-277
Author(s):  
Andrei Alexandru ◽  
Gabriel Ciobanu
Keyword(s):  
Set Theory ◽  
Continuous Functions ◽  
New Order ◽  
Scott Topology ◽  
Topological Properties ◽  
Arbitrary Group ◽  
Topological Approach ◽  
Lattices Of Subgroups ◽  
Group A ◽  
Algebraic Domains

Abstract Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroups lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remains also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.

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