{"id":47,"date":"2012-10-05T13:41:30","date_gmt":"2012-10-05T11:41:30","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=47"},"modified":"2022-11-19T15:35:20","modified_gmt":"2022-11-19T14:35:20","slug":"quasi-polish-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=47","title":{"rendered":"Quasi-polish spaces"},"content":{"rendered":"

Polish spaces are an important class of spaces. \u00a0I am dealing with them in Section 7.7.<\/p>\n

Most notably, they are the kinds of spaces where topological measure theory works best. \u00a0But they also occur naturally in several other situations. \u00a0For example, among all\u00a0T<\/em>3<\/sub>\u00a0spaces, they are exactly those which have \u03c9-continuous\u00a0models<\/em>. \u00a0This is Martin’s first theorem (Theorem 7.7.23 in the book<\/a>), and I’m explaining what this is about in Section 7.7.2.<\/p>\n

I had a conundrum with Polish spaces. \u00a0These are Hausdorff spaces, and the title of the book<\/a> says \u201cnon<\/em>-Hausdorff topology\u201d, right?<\/p>\n

Well, first, what I mean with non-Hausdorff topology is topology that is not restricted to Hausdorff spaces only. \u00a0So I never felt forced to only deal with spaces that are not Hausdorff.<\/p>\n

Still, is there a non-Hausdorff generalization of Polish spaces? \u00a0I have tried to think about it for some time, but could not find anything convincing.<\/p>\n

The news is that\u00a0Matthew de Brecht<\/a>\u00a0managed to find one, which he called\u00a0quasi-Polish<\/em>\u00a0spaces. \u00a0He even went as far as proving the analogs of all the classical theorems on Polish spaces in his new setting. \u00a0(Thanks to\u00a0Klaus Keimel<\/a>, who pointed me to this piece of work.)<\/p>\n

His paper can be found on\u00a0arXiv<\/a>\u00a0[1]. \u00a0Here are a few highlights:<\/p>\n