The Historical Development of Uniform, Proximal, and Nearness Concepts in Topology

Bentley, H. L.; Herrlich, Horst; Hušek, Miroslav

doi:10.1007/978-94-017-1756-4_7

Cited by 28 publications

(11 citation statements)

References 145 publications

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“…There are two more approaches to define uniformity on a set. One of them [18] uses a certain specification of a system of coverings on X. The other is via a system of pseudo-metrics.…”

Section: Preliminariesmentioning

confidence: 99%

A study of uniformities on the space of uniformly continuous mappings

et al. 2020

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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.

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“…There are two more approaches to define uniformity on a set. One of them [18] uses a certain specification of a system of coverings on X. The other is via a system of pseudo-metrics.…”

Section: Preliminariesmentioning

confidence: 99%

A study of uniformities on the space of uniformly continuous mappings

et al. 2020

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“…Remark. There is a strong analogy between the passage from X rig (with its Grothendieck topology defined in terms of rational domains) to 9 the author learned this result just after noticing that the counter-example on p. 55 of [34] is wrong -a fortunate mistake: proving that (along Lambrinos' theorem but contrarily to what is stated in loc. cit.)…”

Section: 4mentioning

confidence: 99%

Uniform Sheaves and Differential Equations

André¹

2012

Rend. Sem. Mat. Univ. Padova

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Real blow-ups and more refined "zooms" play a key role in the analysis of singularities of complex-analytic differential modules. They do not change the underlying topology, but the uniform structure.This suggests to revisit the cohomology theory of differential modules with help of a suitable new notion of uniform sheaves based on the uniformity rather than the topology. We also investigate the p-adic situation (in particular, the analog of real blow-ups) from this uniform viewpoint.

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“…For a history of all kinds of "uniform" type structures, see [7,22]. Besides the three desirable properties mentioned above, Near provides a setting for handling a wide class of extensions of topological spaces.…”

Section: Introductionmentioning

confidence: 98%