From: Brandsma Newsgroups: sci.math Subject: Re: Q: Lusin space (definition) Date: Thu, 14 Jan 1999 13:52:24 +0100 Keywords: What is a Lusin space in topology? ncf@math.uio.no wrote: > Being far away from my library - what is a Lusin space? > (I remember the topological term "crowded space", but I don't know what > _that_ means either...) > > please mail me a copy of the answer (mailto:ncf@math.uio.no if Netscape > messes up my preference file) > > regards, > ncf A Luzin space is a crowded Hausdorff space in which every nowhere dense set is countable. Here Hausdorff is the usual (T_2), crowded means that there are no isolated points, nowhere dense means the closure has non-empty interior. Henno Brandsma ============================================================================== From: Brandsma Newsgroups: sci.math Subject: Re: Q: Lusin space (definition) Date: Fri, 15 Jan 1999 09:40:48 +0100 KRamsay wrote: > >nowhere dense means the closure has non-empty interior. > > Empty interior. Oops, being too quick.. As an addition: it is still open whether these spaces exist in ZFC. It is known that if there is a model of ZFC without L-spaces (regular=T_3, hereditarily Lindelo"f, non-separable spaces) than there can be no Luzin space. Under CH there are plenty of L-spaces eg, and also Luzin spaces. (With Suslin lines, these are the most common type of L-spaces, it is fairly easy to see that Lusin spaces are hereditarily Lindelo"f ( Lindelo"f need not imply T_3 here)). So producing a ZFC Luzin space gives ZFC L-spaces. (most people in topology believe that these do not exist..) Henno