Pretopologies, preuniformities and probabilistic metric spaces |
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Authors: | M Chicourrat CD Horváth |
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Institution: | (1) Département de Mathématiques, Université Blaise Pascal, 63177 Aubiére, France;(2) Département de Mathématiques, Université de Perpignan, 66860 Perpignan, Cedex, France |
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Abstract: | Summary A pretopology on a given set can be generated from a filter of reflexive relations on that set (we call such a structure a
preuniformity). We show that the familly of filters inducing a given pretopology on Xform a complete lattice in the lattice of filters on X. The smallest and largest elements of that lattice are explicitly given. The largest element is characterized by a condition
which is formally equivalent to a property introduced by Knaster--Kuratowski--Mazurkiewicz in their well known proof of Brouwer's
fixed point theorem. Menger spaces and probabilistic metric spaces also generate pretopologies. Semi-uniformities and pretopologies
associated to a possibly nonseparated Menger space are completely characterized. |
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Keywords: | pretopologies preuniformities semi-uniformities Menger spaces probabilistic metric spaces |
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