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  • Author or Editor: M. Chicourrat x
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We answer a question of Császár [12]: under which conditions a given pretopological closure or proximity can be induced by a Cauchy structure? We give a characterization for these closures and proximities using properties of convergences and nasses [14] induced by Cauchy structures. We prove also that the set of Cauchy screens inducing a given reciprocal convergence structure is a non empty interval of the set of Cauchy screens equipped with the usual inclusion order.

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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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