In a Hausdorff but not regular space, collapsing certain closed sets to a point may produce a non-Hausdorff space. Does there exist a term for closed sets one may collapse and still have a Hausdorff space?
Similar question for spaces regular but not normal.
Lacking a better term, let me call such closed sets "nice-1" and "nice-2."
Then one can weaken the notion of compactness by asking merely that finite intersection property families of nice-i closed sets have non-empty intersection (for i=1 or 2). Do either of these weakenings of compactness occur in the literature and/or have a name?