From: zundark1@alberteinstein.co.uk (Zundark)
Subject: Re: regular 2nd countable spaces
Date: 3 Aug 2001 13:43:38 -0700
Newsgroups: sci.math
Summary: More topological separation axioms

William Elliot wrote:

> Any other T_n's beyond T0, 1/4, 1/2, 3/4, 1, 2, 3, 3.5, 4 & 5 ?

Here are a couple more:

  T_{-1}     Every minimal nonempty closed set is a singleton.
  T_{2 1/2}  Distinct points have disjoint closed neighbourhoods.

Also note that some people use T_pi instead of T_{3 1/2},
probably mainly because it looks neater, but also no doubt
because Tychonov spaces feel closer to T_3 than to T_4.

T_2.5 spaces are sometimes called completely Hausdorff spaces
or Urysohn spaces.  (But I've also seen 'Urysohn space' used in
a stronger sense.)

Note that

  X is T_0  <=>  X is hereditarily T_-1.

-- 
Zundark
==============================================================================

From: "Eckertson,Fred" <FECKERTSON@cerner.com>
Subject: Re: regular 2nd countable spaces
Date: Fri, 3 Aug 2001 16:10:11 -0500 
Newsgroups: sci.math

I am accustomed to T_2.5 meaning completely Hausdorff or Urysohn in the
stronger sense, viz., for any two points p != q there is a continuous
function into [0,1] with f(p) = 0 and f(q) = 1. Compare with the
definition of T_3.5 meaning completely regular or Tychonoff.

> -----Original Message-----
[quote of previous message deleted --djr]
==============================================================================

From: zundark1@alberteinstein.co.uk (Zundark)
Subject: Re: regular 2nd countable spaces
Date: 4 Aug 2001 01:19:34 -0700
Newsgroups: sci.math

Eckertson,Fred wrote:

> From: zundark1@alberteinstein.co.uk (Zundark)

> > T_{2 1/2}  Distinct points have disjoint closed neighbourhoods.

> > T_2.5 spaces are sometimes called completely Hausdorff spaces
> > or Urysohn spaces.  (But I've also seen 'Urysohn space' used in
> > a stronger sense.)

> I am accustomed to T_2.5 meaning completely Hausdorff or Urysohn in the
> stronger sense, viz., for any two points p != q there is a continuous
> function into [0,1] with f(p) = 0 and f(q) = 1. Compare with the
> definition of T_3.5 meaning completely regular or Tychonoff.

Yes, I think I got 'completely Hausdorff' and 'Urysohn'
mixed up - 'Urysohn' is usually used in the weaker sense and
'completely Hausdorff' in the stronger sense (although I've
seen them the other way around too).

But T_2.5 has to have the weaker meaning if it is to lie between
T_2 and T_3, and I think this is the only sense I've seen it in.

-- 
Zundark
==============================================================================

From: zundark@thing.co.uk (Zundark)
Subject: Re: Compact set and continous function
Date: 25 Aug 2001 09:27:17 -0700
Newsgroups: sci.math

William Elliot wrote:

> Is KC the most likely candidate for T_(1.5) ?

There are many other properties of interest between T_1 and T_2.
Here are some:

  US  (No sequence has more than one limit.)
  Every Retract Is Closed  (Perhaps this one should be called Eric?)
  T_1 + Sober  (The concept of sober spaces comes from locale theory.)

Some others are discussed in the following paper, which also
defines T_(1/3):

Arenas, F.G., Dontchev, J., Puertas, M.L.,
Unification approach to the separation axioms between
  T_0 and completely Hausdorff.
Acta Math. Hung. 86, No.1-2, 75-82 (2000).

A pre-publication version of the paper is available here:

  http://arxiv.org/abs/math.GN/9810074

-- 
Zundark