Old and new results on quasi-uniform extension

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Abstract

According to [17] or [12], U is a quasi-uniformity on a set X iff it is a filter on X × X, the diagonal Δ = [(x, x): x ∈ X} ⊃ U for U ∈ U (i.e. U is composed of entourages on X), and, for each U ∈ U', there is U'2 U such that U'2 = U'2 o U' = [(x, z): ∃y with (x,s y), (y, z) 2 U'} ⊃ U. The restriction u | X0 to X0 ⊃ X of the quasi-uniformity u on X is composed of the sets U | X0 = U ∩ (X0 × X0) for U ∈ u; it is a quasi-uniformity on X0. Let Y ⊃ X, W be a quasi-uniformity on Y ; W is an extension of the quasi-uniformity u on X if W | X = u. The purpose of the present paper is to give a survey on results, due mainly to Hungarian topologists, concerning extensions of quasi-uniformities.

Original languageEnglish
Pages (from-to)75-85
Number of pages11
JournalRendiconti dell'Istituto di Matematica dell'Universita di Trieste
Volume30
Publication statusPublished - Jan 1 1999

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ASJC Scopus subject areas

  • Mathematics(all)

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