MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.