General topology terminology questions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:05:31Z http://mathoverflow.net/feeds/question/92423 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92423/general-topology-terminology-questions General topology terminology questions David Feldman 2012-03-28T00:05:34Z 2012-03-28T00:22:57Z <p>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?</p> <p>Similar question for spaces regular but not normal.</p> <p>Lacking a better term, let me call such closed sets "nice-1" and "nice-2."</p> <p>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?</p> http://mathoverflow.net/questions/92423/general-topology-terminology-questions/92425#92425 Answer by Ruadhaí Dervan for General topology terminology questions Ruadhaí Dervan 2012-03-28T00:22:57Z 2012-03-28T00:22:57Z <p>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 <a href="http://arxiv.org/abs/math/0512411" rel="nofollow">these notes</a> by Richard Thomas.</p>