Constructible sets in Hausdorff spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:11:37Z http://mathoverflow.net/feeds/question/109007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109007/constructible-sets-in-hausdorff-spaces Constructible sets in Hausdorff spaces Laurent Moret-Bailly 2012-10-06T15:42:54Z 2012-10-25T19:17:24Z <p>In an attempt to write a proof by contradiction, I end up with a space $X$ with the following properties:</p> <p>(0) $X$ is nonempty,<br> (1) $X$ is Hausdorff,<br> (2) $X$ has no isolated points,<br> (3) every subspace of $X$ is constructible (finite union of locally closed subsets).</p> <blockquote> <p>Is this indeed a contradiction? </p> </blockquote> <p>It would suffice to know that any $X$ with properties (1) and (2) has a dense subset with dense complement: such a set cannot be constructible unless $X=\emptyset$. [Edit: there are counterexamples to this, see the comment by Yves Cornulier]</p> http://mathoverflow.net/questions/109007/constructible-sets-in-hausdorff-spaces/110698#110698 Answer by Ramiro de la Vega for Constructible sets in Hausdorff spaces Ramiro de la Vega 2012-10-25T19:17:24Z 2012-10-25T19:17:24Z <p>No, it is not a contradiction.</p> <p>A space $X$ is called <em>submaximal</em> if every subset of $X$ is locally closed (and hence constructible). It is easy to see that spaces with only finitely many non-isolated points are submaximal. Finding a submaximal Hausdorff space with no isolated points seems harder but there are plenty of those too.</p> <p>It is a nice exercise to show that any maximal space is in fact submaximal. A Hausdorff space $(X,\tau)$ is called <em>maximal</em> if it has no isolated points but any finer topology $\tau' \supset \tau$ has isolated points. With a standard use of Zorn´s lemma you can show that any topology without isolated points is contained in a maximal topology. This gives you a way of "constructing" a lot of submaximal Hausdorff spaces with no isolated points. Some of them are even Tychonoff spaces (see E. van Douwen, "<em>Applications of maximal topologies</em>", Topology Appl., 1993), although these are a lot harder to get.</p>