Non-Hausdorff Compactness

From: William Elliot (marsh_at_privacy.net)
Date: 01/15/05 

Date: Sat, 15 Jan 2005 01:46:27 -0800

What terms are used to describe A when

1) some compact K with A subset K
2) cl A compact

I've heard the questionable "relatively compact" for 1).
For 2) I've heard "precompact," which seems somewhat ok.

Have you heard of others? 

-- 
Is there an expression for a space with the property
 	for all compact K, cl K is compact
or have you suggestion?
This property is weaker than kc space, ie all compact sets are closed.
For example, any compact space is compact closure compact, as I'll call
it, such as the cofinite space, which is a T1, not kc space.
How does compact closure compact relate to T1?
An infinite include point topology is T0, not T1
and not compact closure compact.
Is there example of a T1 space that's not compact closure compact?
--
I'm looking into the subtile differences between the two forms of
locally compact
 	for all x, some compact K with x in int K
and strongly locally compact, as Steen and Seebach use,
 	for all x, some compact closed K with x in int K
When a space is compact closure compact, the two are equivalent.
Of all the conditions that imply equivalence, this one seems
the most likely for a converse.
Thank you for what insights you may impart upon this.
----

Relevant Pages

  • Re: Non-Hausdorff Compactness
    ... It's also locally compact. ... what's neat about your example is it's locally compact. ... compact closure compact and not locally compact. ... the distinction vanishes and all such spaces are ...
    (sci.math)
  • Re: Recovering dropped tables
    ... > Using the compact icon in Query Analyzer seems to have no effect. ... > HResult -2147467259 int ... > Message "A duplicate value cannot be inserted into a unique index. ...
    (microsoft.public.sqlserver.ce)
  • Re: struct problems
    ... Clean. ... the person who wrote that might think pj was an int*. ... If you want compact use a typedef. ... Cat kitty1; ...
    (comp.lang.c)
  • Re: First countability
    ... regular S, G_delta, some compact K with x in int K ... is countable local base for x ... or Hausdorff space with all points G_delta is 1st countable. ...
    (sci.math)
  • Re: Closed and compact implies compact intersection
    ... > I think this follows from the fact that: If F is a closed subset of a ... > compact space X then F is compact. ... > Since E is compact in particular is closed hence E int F is closed. ... Jose Carlos Santos ...
    (sci.math)