Atlas: Cauchy completions of uniform frames by Doitchin Doitchinov

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The 1996 Joint Spring Topology Conference and Southeast Dynamical Systems Conference
March 7-9, 1996
Ball State University
Muncie, IN, USA

Organizers
John Emert, Kerry Jones, Roger Nelson

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Cauchy completions of uniform frames
by
Doitchin Doitchinov
Institute of Mathematics, Sofia

For uniform frames the ëntourage" definition, proposed by P.Fletcher and W.Hunsaker in 1991, is used in a slightly modified version.

A uniform frame (L, U) is called Cauchy complete if every Cauchy filter on it is convergent.

A Cauchy complete uniform frame (L^*, U^*) is said to be a Cauchy completion of the uniform frame (L, U) if there exists a dense surjection j: (L^*, U^*) --> (L, U ).

Theorem. Every uniform frame (L, U) possesses a Cauchy completion (L^*, U^*), unique up to an isomorphism and called standard Cauchy completion, together with a dense surjection j: (L^*, U^*) --> (L, U), unique up to the composition with an automorphism and called standard dense surjection, such that:

when the uniform frame (L, U) is Cauchy complete, it is isomorphic with (L^*, U^*);

for any Cauchy completion (L', U') of (L, U) and for any dense surjection j': (L', U') --> (L, U) there is a (unique) dense surjection \psi: (L', U') --> (L^*, U^*) such that j' = j o \psi;

in the spatial case, if the uniform frame (L, U) is generated by the T₀-uniform space (X, U), its standard Cauchy completion (L^*, U^*) is isomorphic with the uniform frame (O [X\tilde], [U\tilde]) generated by the uniform completion ([X\tilde], [U\tilde] ) of (X, U).

Date received: February 14, 1996