scholarly journals On continuous extension of uniformly continuous functions and metrics

2009 ◽  
Vol 116 (2) ◽  
pp. 191-202 ◽  
Author(s):  
T. Banakh ◽  
N. Brodskiy ◽  
I. Stasyuk ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Extension ◽  
Continuous Functions ◽  
Uniformly Continuous

On the Extension of Uniformly Continuous Functions(1)

1975 ◽  
Vol 18 (1) ◽  
pp. 143-145 ◽  
Author(s):  
L. T. Gardner ◽  
P. Milnes
Keyword(s):  
Continuous Function ◽  
Uniform Space ◽  
Continuous Extension ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Totally Bounded ◽  
Special Cases ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

AbstractA theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.

scholarly journals Lattices of uniformly continuous functions

2013 ◽  
Vol 160 (1) ◽  
pp. 50-55 ◽  
Author(s):  
Félix Cabello Sánchez ◽  
Javier Cabello Sánchez
Keyword(s):  
Continuous Functions ◽  
Uniformly Continuous

scholarly journals Compactifications of totally bounded quasi-uniform spaces

1986 ◽  
Vol 28 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P. Fletcher ◽  
W. F. Lindgren
Keyword(s):  
Hausdorff Space ◽  
Uniform Space ◽  
Continuous Extension ◽  
Sufficient Condition ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Totally Bounded ◽  
Continuous Map ◽  
Total Boundedness ◽  
Uniformly Continuous

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

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