Lower semifinite topology in hyperspaces

Abstract

In this paper we use the lower semifinite topology in hyperspaces to obtain examples in topology such as pseudocompact spaces not countably compact, separable spaces not Lindelöf and in a natural way many spaces appear which are T0 but not T1. We also give, in a unified form, many examples of contractible locally contractible spaces, absolute extensor for the class of all topological spaces, absolute retract and many examples of spaces having the fixed point property. Finally we obtain the following characterization of compactness: ``A paracompact Hausdorff space is compact if and only if the hyperspace, with the lower semifinite topology, 2X, has the fixed point property''.