LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

2020 ◽  
Vol 4 (1) ◽  
pp. 29-39
Author(s):  
Dilrabo Eshkobilova ◽  
Keyword(s):  
Compact Support ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Continuous Maps ◽  
Uniformly Continuous

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

Categorical characterization of uniform hyperspaces

1983 ◽  
Vol 94 (2) ◽  
pp. 229-233 ◽  
Author(s):  
J. M. Harvey
Keyword(s):  
Uniform Spaces ◽  
Categorical Methods ◽  
Continuous Maps ◽  
Uniformly Continuous

In this paper, we use categorical methods to arrive at a characterization of Hausdorff hyperspaces of separated uniform spaces within a category of ordered uniform spaces. We begin by showing that the hyperspace construction gives rise to a monad H in the category Uni of non-empty separated uniform spaces and uniformly continuous maps. We then identify the category of H-algebras and characterize hyperspaces in this context.

scholarly journals A study of uniformities on the space of uniformly continuous mappings

2020 ◽  
Vol 18 (1) ◽  
pp. 1478-1490
Author(s):  
Ankit Gupta ◽  
Abdulkareem Saleh Hamarsheh ◽  
Ratna Dev Sarma ◽  
Reny George
Keyword(s):  
Uniform Space ◽  
Uniform Spaces ◽  
Continuous Mappings ◽  
Uniformly Continuous

Abstract New families of uniformities are introduced on UC(X,Y) , the class of uniformly continuous mappings between X and Y, where (X,{\mathcal{U}}) and (Y,{\mathcal{V}}) are uniform spaces. Admissibility and splittingness are introduced and investigated for such uniformities. Net theory is developed to provide characterizations of admissibility and splittingness of these spaces. It is shown that the point-entourage uniform space is splitting while the entourage-entourage uniform space is admissible.

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].

Spaces with a Unique Uniformity

1983 ◽  
Vol 35 (6) ◽  
pp. 1001-1009
Author(s):  
Richard H. Warren
Keyword(s):  
Companion Paper ◽  
The Other ◽  
Completely Regular ◽  
Continuous Maps ◽  
Uniformly Continuous ◽  
Compact Subsets

The major results in this paper are nine characterizations of completely regular spaces with a unique compatible uniformity. All prior results of this type assumed that the space is Tychonoff (i.e., completely regular and Hausdorff) until the appearance of a companion paper [9] which began this study. The more important characterizations use quasi-uniqueness of R1-compactifications which relate to uniqueness of T2-comPactifications. The features of the other characterizations are: (i) compact subsets linked to Cauchy filters, (ii) C- and C*-embeddings, and (iii) lifting continuous maps to uniformly continuous maps.Section 2 contains information on T0-identification spaces which we will use later in the paper. In Section 3 several properties of uniform identification spaces are developed so that they can be used later. The nine characterizations are established in Section 4. Also it is shown that a space with a unique compatible uniformity is normal if and only if each of its closed subspaces has a unique compatible uniformity.

scholarly journals Paracompact-type mappings

2021 ◽  
Vol 102 (2) ◽  
pp. 62-66
Author(s):  
B.E. Kanetov ◽  
◽  
A.M. Baidzhuranova ◽  
Keyword(s):  
Uniform Space ◽  
Continuous Mapping ◽  
Uniform Spaces ◽  
Uniform Topology ◽  
Point Space ◽  
Continuous Mappings ◽  
Basic Concepts ◽  
Uniformly Continuous

Recently a new direction of uniform topology called the uniform topology of uniformly continuous mappings has begun to develop intensively. This direction is devoted, first of all, to the extension to uniformly continuous mappings of the basic concepts and statements concerning uniform spaces. In this case a uniform space is understood as the simplest uniformly continuous mapping of this uniform space into a one-point space. The investigations carried out have revealed large uniform analogs of continuous mappings and made it possible to transfer to uniformly continuous mappings many of the main statements of the uniform topology of spaces. The method of transferring results from spaces to mappings makes it possible to generalize many results. Therefore, the problem of extending some concepts and statements concerning uniform spaces to uniformly continuous mappings is urgent. In this article, we introduce and study uniformly R-paracompact, strongly uniformly R-paracompact, and uniformly R-superparacompact mappings. In particular, we solve the problem of preserving R-paracompact (respectively, strongly uniformly R-paracompact, uniformly R-superparacompact) spaces towards the preimage under uniformly R-paracompact (respectively, strongly uniformly R-paracompact, uniformly R-superparacompact) mappings.

scholarly journals Connecting ergodicity and dimension in dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 451-462 ◽  
Author(s):  
C. D. Cutler
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Pointwise Dimension ◽  
Continuous Maps ◽  
Borel Probability ◽  
The Relationship

AbstractIn this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

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.

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