Spaces for which all intermediate spaces are Wallman equivalent (Q808468)

For a \(T_ 1\) space X, let WX denote its Wallman compactification and \(e_ X\) the canonical embedding \(X\to WX\). If \(f:X\to Y\) is continuous, a Wallman extension of f (which may not exist) is a continuous map \(f^*: WX\to WY\) such that \(e_ Y\circ f=f^*\circ e_ X\). The main result reads: A space X has the property that, for every Y such that \(X\subset Y\subset WX\), the embedding \(\iota\) : \(X\to Y\) has a Wallman extension \(\iota^*: WX\to WY\) which is a homeomorphism, if and only if whenever \(X\subset Y\subset WX\) and A, B disjoint closed in Y, the closures in WX of A and B are disjoint. Such spaces X are called WSM (Wallman separation maintaining). Some of their properties are studied.