Pseudo-uniformities
Tullio Valent
Università di Padova, Italy
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Abstract
The notions of pseudo-uniformity for a set and of pseudo-metric presented in this paper are somehow connected. Unlike the uniformity, the elements of a pseudo-uniformity do not necessarily contain the diagonal of but only some points (at least one point) of it. Likewise, the pseudo-metrics considered and utilized here do not vanish on the whole of , but only in some points of it. It will be showed how any family of pseudo-metrics defines a pseudo-uniformity. The main result achieved is the proof that every pseudo-uniformity is defined by a family of “pseudo-uniformly continuous” pseudo-metrics. In the final part of the paper, we prove that every topology is defined by a family of pseudo-metrics and, consequently, that every topology is pseudo-uniformizable: nay every topology defines a pseudo-uniformity which induces just the starting topology.
Cite this article
Tullio Valent, Pseudo-uniformities. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 34 (2023), no. 1, pp. 89–99
DOI 10.4171/RLM/998