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