Constructions and applications of rigid spaces, I

Abstract

The two major results proved are: (1) The category TOP of topological spaces contains a complete nonreflective subcategory. (2) Under the assumption (2^m)⁺ < 2^2m, for each infinite cardinal number m there exists a Hausdorff space of cardinality m, in which the identity map is the only nonconstant continuous self-map. The first result is proved as a consequence of another result which answers a question of Herrlich concerning strongly rigid spaces; it is then used to settle in the negative a conjecture concerning the characterization of reflective subcategories in TOP. In addition, several interesting spaces are constructed.