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T 2 and T 3 objects at p in the category of proximity spaces

Muammer Kula, Samed Özkan (2020)

Mathematica Bohemica

In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point p in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various Pre T 2 , T i , i = 0 , 1 , 2 , 3 , structures at a point p are investigated. Finally, we examine the relationships between the generalized separation...

The minimum uniform compactification of a metric space

R. Grant Woods (1995)

Fundamenta Mathematicae

It is shown that associated with each metric space (X,d) there is a compactification u d X of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of u d X are presented, and a detailed study of the structure of u d X is undertaken. This culminates in a topological characterization of the outgrowth u d n n , where ( n , d ) is Euclidean n-space with its usual metric.

Totally bounded frame quasi-uniformities

Peter Fletcher, Worthen N. Hunsaker, William F. Lindgren (1993)

Commentationes Mathematicae Universitatis Carolinae

This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity on a frame L there is a totally bounded quasi-uniformity on L that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines . The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum ψ L and the compactification L of a uniform frame ( L , 𝐔 ) are meaningful for quasi-uniform frames. If 𝐔 is a totally bounded quasi-uniformity...

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