On Point-finiteness in Pointfree Topology

Abstract

In pointfree topology, the point-finite covers introduced by Dowker and Strauss do not behave similarly to their classical counterparts with respect to transitive quasi-uniformities, contrarily to what happens with other familiar types of interior-preserving covers. The purpose of this paper is to remedy this by modifying the definition of Dowker and Strauss. We present arguments to justify that this modification turns out to be the right pointfree definition of point-finiteness. Along the way we place point-finite covers among the classes of interior-preserving and closure-preserving families of covers that are relevant for the theory of (transitive) quasi-uniformities, completing the study initiated with [6].

The Katětov-Morita theorem for the dimension of metric frames

“…In particular we thank the referee for alerting us to the paper by Ferreira and Picado (see [6]), which to our mind as well, has the right pointfree definition of point-finiteness.…”

“…It should be further remarked that the point-finite notion of Dowker and Strauss has some shortcomings mainly because it does not behave so well as its classical counterpart. In a recent article by Ferreira and Picado (see [6]), this deficiency was corrected. These authors proposed a stronger version of point-finiteness, still weaker than local finiteness however.…”

Notes on Exact Meets and Joins

“…tcpaq | a Au is closure-preserving). Interior-preserving covers play a decisive role in the construction of canonical examples of transitive quasi-uniformities for frames ( [10]). Of course, any interior-preserving cover of L is closure-preserving but, somewhat surprisingly and contrary to what happens in spaces, the converse does not hold in general.…”