An Algebraic Characterization of Remainders of Compactifications

1983 ◽  
Vol 26 (3) ◽  
pp. 347-350
Author(s):  
James Hatzenbuhler ◽  
Don A. Mattson ◽  
Walter S. Sizer
Keyword(s):  
Hausdorff Space ◽  
Metric Spaces ◽  
Quotient Ring ◽  
Algebraic Characterization ◽  
Locally Compact ◽  
Completely Regular ◽  
Compact Metric Spaces ◽  
Primitive Idempotents

AbstractLet X be a locally compact, completely regular Hausdorff space. In this paper it is shown that all compact metric spaces are remainders of X if and only if the quotient ring C*(X)/C∞(X) contains a subring having no primitive idempotents.

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

1972 ◽  
Vol 24 (4) ◽  
pp. 622-630 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Locally Compact ◽  
Completely Regular ◽  
Isolated Points ◽  
Remote Points ◽  
Topological Boundary ◽  
Regular Closed

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

scholarly journals On Hausdorff compactifications of non-locally compact spaces

1979 ◽  
Vol 2 (3) ◽  
pp. 481-486
Author(s):  
James Hatzenbuhler ◽  
Don A. Mattson
Keyword(s):  
Hausdorff Space ◽  
Locally Compact ◽  
Completely Regular ◽  
Compact Spaces ◽  
Hausdorff Compactifications ◽  
Set Of Points ◽  
Locally Compact Spaces

LetXbe a completely regular, Hausdorff space and letRbe the set of points inXwhich do not possess compact neighborhoods. AssumeRis compact. IfXhas a compactification with a countable remainder, then so does the quotientX/R, and a countable compactificatlon ofX/Rimplies one forX−R. A characterization of whenX/Rhas a compactification with a countable remainder is obtained. Examples show that the above implications cannot be reversed.

Simple Links in Locally Compact Connected Hausdorff Spaces are Nondegenerate

1981 ◽  
Vol 34 (2) ◽  
pp. 349-355
Author(s):  
David John
Keyword(s):  
Hausdorff Space ◽  
Metric Spaces ◽  
Connected Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Peano Continua

The fact that simple links in locally compact connected metric spaces are nondegenerate was probably first established by C. Kuratowski and G. T. Whyburn in [2], where it is proved for Peano continua. J. L. Kelley in [3] established it for arbitrary metric continua, and A. D. Wallace extended the theorem to Hausdorff continua in [4]. In [1], B. Lehman proved this theorem for locally compact, locally connected Hausdorff spaces. We will show that the locally connected property is not necessary.A continuum is a compact connected Hausdorff space. For any two points a and b of a connected space M, E(a, b) denotes the set of all points of M which separate a from b in M. The interval ab of M is the set E(a, b) ∪ {a, b}.

scholarly journals Quantum locally compact metric spaces

2013 ◽  
Vol 264 (1) ◽  
pp. 362-402 ◽  
Author(s):  
Frédéric Latrémolière
Keyword(s):  
Metric Spaces ◽  
Locally Compact ◽  
Compact Metric Spaces

scholarly journals Multipliers of Modules of Continuous Vector-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liaqat Ali Khan ◽  
Saud M. Alsulami
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Normed Spaces ◽  
Locally Compact ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space ◽  
Locally Convex

In 1961, Wang showed that ifAis the commutativeC*-algebraC0(X)withXa locally compact Hausdorff space, thenM(C0(X))≅Cb(X). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing thatHomC0(X,A)(C0(X,E),C0(X,F))≃Cs,b(X,HomA(E,F)),whereEandFarep-normed spaces which are also essential isometric leftA-modules withAbeing a certain commutativeF-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

scholarly journals WEIGHTED COMPOSITION OPERATORS AND DYNAMICAL SYSTEMS

1998 ◽  
Vol 29 (2) ◽  
pp. 101-107
Author(s):  
R. K. SINGH ◽  
BHOPINDER SINGH
Keyword(s):  
Hausdorff Space ◽  
Weighted Space ◽  
Composition Operators ◽  
Continuous Functions ◽  
Weighted Composition Operators ◽  
Linear Dynamical System ◽  
Completely Regular ◽  
Weighted Composition ◽  
Locally Convex

Let $X$ be a completely regular Hausdorff space, $E$ a Hausdorff locally convex topo­logical vector space, and $V$ a system of weights on $X$. Denote by $CV_b(X, E)$ ($CV_o(X, E)$) the weighted space of all continuous functions $f : X \to E$ such that $vf (X)$ is bounded in $E$ (respectively, $vf$ vanishes at infinity on $X$) for all $v \in V$. In this paper, we give an improved characterization of weighted composition operators on $CV_b(X, E)$ and present a linear dynamical system induced by composition operators on the metrizable weighted space $CV_o(\mathbb{R}, E)$.

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