A Galois view of uniformities
by
Maria João Ferreira
Departamento de Matemática, Universidade de Coimbra
Coauthors: Jorge Picado (Departamento de Matemática, Universidade de Coimbra)

In [1] P. Fletcher, W. Hunsaker and W. Lindgren proved that every (quasi-) uniformity U on a set X is characterized by the collection of all maps

E exists: T(U) --> T(U) A --> {b in X | existsa in A  (a, b) in E} (E in B)

on the topology T( U) determined by U, for some base B of U. On the other hand, U may also be characterized by the collection of all maps

E for all: P(X) --> P(X) A --> {b in X | for alla in A  (a, b) in E} (E in B).

Each E _exists is a residuated map (that is, a covariant Galois connection) on the frame T( U). On the other hand, each E _{for all} is a (contravariant) Galois connection on P(X).

In this talk we show that this is embodied by a general situation in pointfree topology and that the equivalence between these two descriptions is a consequence of a general phenomenon on Galois connections in frames. We also show how to replace maps E _{for all} by Galois maps completely defined on the frame T(U).

[1] P. Fletcher, W. Hunsaker and W. Lindgren, A functional approach to uniform and quasi-uniform frames, in: Topology with Applications, Bolyai Soc. Math. Studies, Vol. 4, 1993, p. 217-222.

Date received: March 2, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-58.