Re: Locally Compact Subspaces

On Fri, 09 Mar 2007 10:25:39 +0000, José Carlos Santos
<jcsantos@xxxxxxxx> wrote:

On 09-03-2007 10:08, freensway@xxxxxxxxxxx wrote:

Define a subspace of the real line that can be represented as the
union of two locally compact subspaces, one of which is closed and the
other open, and that this is not a locally compact space.
Just to provide a definition: A topological space X is called a
locally compact space if for every x in X there exists a neighborhood
U of the poitn x such that the closure of U is a compact subspace of X.
Take A = (0,1/2) U (1/3,1/4) U (1/5,1/6) U ... and B = {1}. Then A is
open, B is closed, A and B are locally compact, but their union isn't.

Thanks. But in general is it just taking any open set that's locally
compact with a singleton sufficiently far from your open set that
yields this result? The problem arises when you union the singleton
with everything else?

My singleton was most definitely *not* far way from my open set. The
point of the singleton is located at the _boundary_ of my open set.

Actually what you wrote implies that A = (0,1/2) and B = {1}.
I suspect that there were typos in the definition of A _and_ the
definition of B - probably if you gave the definition you meant
then these most definite properties would be more clear...

But, of cours, the answer to your first question is negative. If
A = (0,1) and B = {x}, then the union of A and B is locally compact, no
matter where _x_ is.

I do not understand your second question.

Best regards,

Jose Carlos Santos


************************

David C. Ullrich
.

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