Absolutely closed maps

Abstract:

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