Hausdorff Distance and a Compactness Criterion for Continuous Functions

1986 ◽  
Vol 29 (4) ◽  
pp. 463-468 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Uniform Convergence ◽  
Hausdorff Distance ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Criterion ◽  
Uniformly Continuous

AbstractLet 〈X, dx〉 and 〈Y, dY〉 be metric spaces and let hp denote Hausdorff distance in X x Y induced by the metric p on X x Y given by p[(x1, y1), (x2, y2)] = max ﹛dx(x1, x2),dY(y1, y2)﹜- Using the fact that hp when restricted to the uniformly continuous functions from X to Y induces the topology of uniform convergence, we exhibit a natural compactness criterion for C(X, Y) when X is compact and Y is complete.

scholarly journals On a Durrmeyer-type modification of the Exponential sampling series

2020 ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini
Keyword(s):  
Asymptotic Formula ◽  
Uniform Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Convergence Properties ◽  
Sampling Series ◽  
Quantitative Results ◽  
Uniformly Continuous

Abstract In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

Regular completions of uniform convergence spaces

1974 ◽  
Vol 11 (3) ◽  
pp. 413-424 ◽  
Author(s):  
R.J. Gazik ◽  
D.C. Kent ◽  
G.D. Richardson
Keyword(s):  
Uniform Convergence ◽  
Real Line ◽  
Dual Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Convergence Space ◽  
Uniform Spaces ◽  
Convergence Spaces ◽  
Uniform Convergence Space ◽  
Uniformly Continuous

A regular completion with universal property is obtained for each member of the class of u–embedded uniform convergence spaces, a class which includes the Hausdorff uniform spaces. This completion is obtained by embedding each u-embedded uniform convergence space (X, I) into the dual space of a complete function algebra composed of the uniformly continuous functions from (X, I) into the real line.

scholarly journals Slowly oscillating continuity in abstract metric spaces

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 925-930 ◽  
Author(s):  
Hüseyin Çakallı ◽  
Ayșe Sӧnmez
Keyword(s):  
Vector Space ◽  
Compact Subset ◽  
Topological Vector Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Cone Metric Space ◽  
Cone Metric Spaces ◽  
Uniformly Continuous ◽  
Cone Metric

In this paper, we investigate slowly oscillating continuity in cone metric spaces. It turns out that the set of slowly oscillating continuous functions is equal to the set of uniformly continuous functions on a slowly oscillating compact subset of a topological vector space valued cone metric space.

On Uniform Convergence of Continuous Functions and Topological Convergence of Sets

1983 ◽  
Vol 26 (4) ◽  
pp. 418-424 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Related Result ◽  
Continuous Functions ◽  
Topological Convergence ◽  
The Relationship ◽  
Convergence Of Sets

AbstractLet X and Y be metric spaces. This paper considers the relationship between uniform convergence in C(X, Y) and topological convergence of functions in C(X, Y), viewed as subsets of X×Y. In general, uniform convergence in C(X, Y) implies Hausdorff metric convergence which, in turn, implies topological convergence, but if X and Y are compact, then all three notions are equivalent. If C([0, 1], Y) is nontrivial arid topological convergence in C(X, Y) implies uniform converger ce then X is compact. Theorem: Let X be compact and Y be loyally compact but noncompact. Then topological convergence in C(X, Y) implies uniform convergence if and only if X has finitely many components. We also sharpen a related result of Naimpally.

scholarly journals U(X) as a ring for metric spaces X

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1981-1984 ◽  
Author(s):  
Sànchez Cabello
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Infinite Subset ◽  
Short Paper ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

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