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Re: topology question
Posted:
Aug 24, 2006 11:03 AM
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On 23 Aug 2006 08:42:50 -0700, "Snis Pilbor" <snispilbor@yahoo.com>
wrote:
>It's well known there are topologies which are connected but not path
>connected. But can the same be true if the space has only finitely
>many points? It seems to me that in the finite case, connected ought
>to imply simply connected, but I can't prove it.
>
"Simply connected" isn't the same as "path connected" - it is
something more complicated ... (but it *implies* path conn., at
least with the more frequently used definition which explicitely
says so)
>
>Or maybe there's a
>counterexample..
>
I hope you realize that finite *non-discrete* spaces are not
Hausdorff, yes do not even satisfy separation axiom T1
(equivalent to: singetons are closed), so that it is not so easy
to understand them intuitively in the usual manner.
(If the space *is* discrete *and* connected, then it cannot contain
more than 1 point, which implies it is path connected. But this is
rather ... trivial.)
For the non-Hausdorff (non-T1) case, it will take me more time
to give an answer. May-be someone else has given one.
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