From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Topology Questions
Date: 2 Nov 1996 05:43:54 GMT

In article <55e948$1850@pulp.ucs.ualberta.ca>,
U Lange <lange@gpu5.srv.ualberta.ca> wrote:
>Robert Burke Osburn (rosburn@tiger4.ocs.lsu.edu) wrote:
>: I am stumped on these two problems.
>:  
>: (1)  
>:   IS [0,1] (subset of the real line) compact in the following topology, 
>: T: 
>:  T={A: R-A is countable or is all of R}
>: 
>: I can show that [0,1] is compact in the lower limit topology (it is isn't 
>: it?), but I'm not sure about this one.
>
>My topology is a bit rusty, but doesn't
>
>S is a compact set ==> S is a closed set 
>
>hold for any topology? 

Compactness is a property held (or not) by a topological space. Closedness
is a property held (or not) by _pairs_, that is, one must ask if  S  is
closed _in X_  where  X  is  some larger topological space. Sure, every
compact set  S  is _closed in  S_, but then, that's true of any space  S,
compact or not.

It _is_ true that a compact space  S  is closed in any larger topologcal
space  X  which is in addition  Hausdorff. But without Hausdorffness, 
closedness does not necessarily follow (consider  X  finite with
coarse topology,  0 < S < X).

Incidentally, you must be careful here: some authors, including Bourbaki,
define compact spaces as _Hausdorff_ spaces all of whose open covers have
finite subcovers. Surely this is the most common setting anyway, but
the choice of definition means that one theorem or another will have to
be stated with extra conditions. 

Note to the original poster: since you're giving  [0,1]  a new topology
not related in any way to ordering, it is equivalent up to isomorphism
(and perhaps less confusing) to replace  [0,1]  in the problem with any
other set of the same cardinality, such as [0,1] union {2, 3, 4, ...}. 
(Not that cardinality matters here beyond being uncountable...)

>: (2)
>: Given two topolgies T and T' on X where T is contained in T'. What can 
>: you say about the compactness of X in T if X is compact in T'? Also what 
>: can you can about the compactness of X in T' if X is compact in T? 

The "right" way to consider questions like this is not to think about
what sets are in one topology or another. Much better is to think of
(X,T)  and  (X,T')  as two distinct spaces which come equipped with a
very natural one-to-one and onto map between them, namely the identity
map on  X.  In which direction(s) is this map continuous?

Bonus reflection in this direction: if the usual topology on  [0,1]  is
replaced by anything stronger, it's no longer compact. If it's replaced
by anything weaker, it's no longer Hausdorff. Kinda makes you realize
just how special a compact Hausdorff topology is.

By the way, does it bother anyone else to talk about "open covers" and
"finite subcovers"? In the one case, the adjective applies to the cover
itself (as a set (of sets)) and in the other case, it applies to the
elements of that set (of sets).

dave