From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: bijection = homeomorphism
Date: 16 Feb 1995 16:54:04 GMT
In article <3hvfq4$d53@newstand.syr.edu>,
Thomas R. Scavo wrote:
>>> Under what conditions is a continuous bijection a homeomorphism?
>
>... I want to be able to say "If a
>bijection f: X -> Y is continuous and ..., then its inverse is
>continuous." ...
>Isn't it true that X compact will suffice?
No.
If you have a bijection between the points in two spaces, you might as
well think of f as being a continuous map between two topological
spaces built on the same underlying set, that is, view f as
f : (X, T1) --> (X, T2)
where T1 and T2 are two topologies and f is the identity map.
When is f continuous? If and only if T1 contains T2. When is its
inverse continuous? If and only if T2 contains T1. So your question
asks, under what conditions can I assert that two given topologies on a
space are equal, knowing the first contains the second? (Equivalently,
in what family of conditions is the topology T1 minimal?)
Having T1 compact is not enough -- just take T2 = { {}, X } (the
coarse topology).
Having T1 compact and T2 Hausdorff _does_ turn out to be sufficient,
that is, a compact cannot properly contain any Hausdorff topologies.
I find this to be an entertaining perspective: it sort of says compact
topologies have to have relatively few open sets in them. Turned
the other way ("a Hausdorff topology cannot be properly contained in
a compact topology") it says Hausdorff topologies have to have kind of
a lot of open sets. Taken together ("no two compact, Hausdorff topologies
are comparable") this makes it clear that compact+Hausdorff is quite a
restrictive condition, and thus one from which many nice results can
be expected to follow (indeed, Bourbaki _defines_ compact to include
the Hausdorff axiom). In view of this it is a pleasant accident of
Nature that such familiar spaces as [0,1] are indeed compact and Hausdorff.
dave
==============================================================================
Date: Mon, 20 Feb 1995 13:58:49 +1000
To: rusin@washington.math.niu.edu (Dave Rusin)
From: ibokor@metz.une.edu.au (ibokor)
Subject: Re: Re: bijection = homeomorphism
Subject: Re: bijection = homeomorphism
From: Dave Rusin, rusin@washington.math.niu.edu
Date: 16 Feb 1995 16:54:04 GMT
In article <3hvvvc$dhp@watson.math.niu.edu> Dave Rusin,
rusin@washington.math.niu.edu writes:
>In article <3hvfq4$d53@newstand.syr.edu>,
>Thomas R. Scavo wrote:
>>>> Under what conditions is a continuous bijection a homeomorphism?
>>
>>... I want to be able to say "If a
>>bijection f: X -> Y is continuous and ..., then its inverse is
>>continuous." ...
>>Isn't it true that X compact will suffice?
>
>No.
>
>If you have a bijection between the points in two spaces, you might as
>well think of f as being a continuous map between two topological
>spaces built on the same underlying set, that is, view f as
> f : (X, T1) --> (X, T2)
>where T1 and T2 are two topologies and f is the identity map.
>When is f continuous? If and only if T1 contains T2. When is its
>inverse continuous? If and only if T2 contains T1. So your question
>asks, under what conditions can I assert that two given topologies on a
>space are equal, knowing the first contains the second? (Equivalently,
>in what family of conditions is the topology T1 minimal?)
>
>Having T1 compact is not enough -- just take T2 = { {}, X } (the
>coarse topology).
>
>Having T1 compact and T2 Hausdorff _does_ turn out to be sufficient,
>that is, a compact cannot properly contain any Hausdorff topologies.
>I find this to be an entertaining perspective: it sort of says compact
>topologies have to have relatively few open sets in them. Turned
>the other way ("a Hausdorff topology cannot be properly contained in
>a compact topology") it says Hausdorff topologies have to have kind of
>a lot of open sets. Taken together ("no two compact, Hausdorff topologies
>are comparable") this makes it clear that compact+Hausdorff is quite a
>restrictive condition, and thus one from which many nice results can
>be expected to follow (indeed, Bourbaki _defines_ compact to include
>the Hausdorff axiom). In view of this it is a pleasant accident of
>Nature that such familiar spaces as [0,1] are indeed compact and Hausdorff.
>
>dave
Strange! What about the compactification of a non-compact space?
How does that fit into this scheme?
==============================================================================
Date: Mon, 20 Feb 95 09:20:15 CST
From: rusin (Dave Rusin)
To: ibokor@metz.une.edu.au
Subject: Re: Re: bijection = homeomorphism
>>Having T1 compact and T2 Hausdorff _does_ turn out to be sufficient,
>>that is, a compact cannot properly contain any Hausdorff topologies.
>>I find this to be an entertaining perspective: it sort of says compact
>>topologies have to have relatively few open sets in them. Turned
>>the other way ("a Hausdorff topology cannot be properly contained in
>>a compact topology") it says Hausdorff topologies have to have kind of
>>a lot of open sets. Taken together ("no two compact, Hausdorff topologies
>>are comparable") this makes it clear that compact+Hausdorff is quite a
>>restrictive condition, and thus one from which many nice results can
>>be expected to follow (indeed, Bourbaki _defines_ compact to include
>>the Hausdorff axiom). In view of this it is a pleasant accident of
>>Nature that such familiar spaces as [0,1] are indeed compact and Hausdorff.
>>
>>dave
>
>Strange! What about the compactification of a non-compact space?
>How does that fit into this scheme?
It's interesting that you would think of that. A "compactification" of a
space X usually means a larger space Z containing X such that (1) Z is
compact, and (2) the topology of Z agrees with that of X on the subset
X of Z. (Is that clear? You're not supposed to take X={0,1} with the
coarse topology and Z={0,1,2} with the discrete topology, since then when
you view X as a subset of Z, it will have different open sets than the
topological space X did in the first place.)
The key point here is that to make the compactification you add _points_, not
_open sets_. In the discussion you quoted, what stays fixed is the collection
of points; what is varying is the collections of open sets on that set of
points. It follows as a part of that discussion that if (X,T) is a topological
space which is compact, then so is (X, T') where T' is any smaller
(i.e. "coarser") topology on X; thus it would make perfect sense to define
the "Ibokor compactification" of any topological space (X,T) to be the
following space: the collection of points X is not changed, but the
topology is the largest ("finest", or "strongest") one in T in which X is
compact -- the point of all the previous discussion being that this toplogy
is well defined. Undoubtedly there are non-trivial topologies on spaces X
for which this Ibokor compactification would turn out to be the extreme case
(X, coarse). I don't even know, for example, what the I.c. of the open
interval (0,1) would be.
For the record: there are different possible compactifications in the usual
sense, too. There is the one-point compactification X* = X union one more
point, with a certain topology. There is also the Stone-Cech compactification,
which is quite large but of particular importance in functional analysis.
But these all meet the description of Z I posed earlier. I don't know
what Bourbaki has to say about these -- is that what you meant?
dave