Take the 2-minute tour ×  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?
share|improve this question  1 Answer 1  In Geometric Invariant
Theory, the study of quotients in algebraic geometry, some
points are ignored in the quotient (by its construction) that
would make the quotient non-Hausdorff. These points are called
'unstable'. Sometimes the set of all unstable points is called
the 'unstable locus'. This is of course just a special case of
your question, in a slight