Absolutely closed maps

Abstract

An example is given of a continuous function f : X → Y f:X \to Y which is closed, has point inverses H H -closed, but which can be extended to a continuous function F : Z → Y F:Z \to Y for some Z Z which has X X as a proper dense subset. A partial characterization of nonextendable functions is given in terms similar to Bourbaki’s theorem that perfect maps f : X → Y f:X \to Y are those for which f × i Z : X × Z → Y × Z f \times {i_Z}:X \times Z \to Y \times Z is a closed map for all spaces Z Z .