Lattice Theoretic Properties of Quasi-Uniform Extensions
by
Attila Losonczi

The main problem is the following one: Let X, Y be sets, X subset Y, and a quasi-uniformity U and a topology \tau' be given on X and Y respectively, such that \tau'|_X = U^tp where U^tp is the topology induced by U. We are looking for a quasi-uniformity V on Y such that V is an extension of U (i.e. V|_X = U) and V^tp = \tau'. We say that V is a (U, \tau')-compatible extension. When writing (X, U, Y, \tau') we mean that X, Y are sets, X subset Y, and U is a quasi-uniformity on X, \tau' is a topology on Y such that \tau'|_X = U^tp.

The most important questions are:

1. Find necessary and sufficient conditions for the existence of a (U, \tau')-compatible extension.
2. Describe all (U, \tau')-compatible extensions.
3. Investigate the class of all (U, \tau')-compatible extensions.

Here we will discuss the third question only.

The set of all (U, \tau')-compatible extensions is closed for arbitrary suprema hence it has a finest element. But in general it is not a lattice so there is no coarsest extension. Moreover in the following special case of the mentioned problem, the considered extensions do not constitute a lattice: let us assume that a quasi-uniformity W is also given on Y-X such that W^tp = \tau'|_Y-X and we are looking for a quasi-uniformity V on Y such that V|_X = U, V|_Y-X = W, V^tp = \tau'.

When Y-X is finite then the set of all (U, \tau')-compatible extensions is a sup-distributive lattice, in other words it is a frame and inf{V₁, V₂} equals to V₁ \cap V₂, where V_i is a (U, \tau')-compatible extension (i = 1, 2). In the finite case there is no coarsest extension in general. It is true that for every finite distributive lattice there is a system (X, U, Y, \tau') such that the lattice composed of all (U, \tau')-compatible extensions is isomorphic with the given lattice, and for every cardinal number \alpha there is a system (X, U, Y, \tau') such that the cardinality of the set of all (U, \tau')-compatible extensions equals to \alpha.

Let (X, U, Y, \tau') be given where Y-X is finite. Denote by H the lattice of all (U, \tau')-compatible extensions and by H_p the lattice of all (U , \tau'|_{X \cup {p}})-compatible extensions where p in Y-X. If \tau' is a strict extension of \tau = U^tp and \tau'|_Y-X is T₁ then H is the direct sum of H_p-s (p in Y-X).

Some interesting open problems:

1. Is it true that if H is a frame then there is a system (X, U, Y, \tau') such that the lattice composed of all (U, \tau')-compatible extensions is isomorphic with H?
2. Characterize when this lattice is finite or when it has a coarsest element.

1 Á. Császár Extensions of quasi-uniformities Acta Math. Acad. Sci. Hungar. 37 1981 121-145
2 J. Deák Survey of compatible extensions Topology theory and applications II., Colloq. Math. Soc. J. Bolyai
3 A. Losonczi Finite quasi-uniform extensions I

Date received: June 24, 1996

Copyright © 1996 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 # caai-67.