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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?

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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 slightly different area, but perhaps the terminology is used elsewhere. A good reference for this if you're interested is these notes by Richard Thomas.

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