Re: Homeogeneous Spaces

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 01/24/05 

Date: Mon, 24 Jan 2005 05:57:36 -0600

On Sun, 23 Jan 2005 23:09:52 -0800, William Elliot <marsh@privacy.net>
wrote:

>From: David C. Ullrich <ullrich@math.okstate.edu>
>Newsgroups: sci.math
>Subject: Re: Homeogeneous Spaces
>
>> William Elliot <marsh@privacy.net> wrote:
>>> Topological space S is homogeneous when for all x,y in S,
>>> some auto-homeomorphism h:S -> S with h(x) = y.
>>
>>> Is a connected subspace of homogeneous space homogeneous?
>>> No. [0,1] and [0,1) subset R are counterexamples.
>>
>>> Is an open subspace of homogeneous space homogeneous?
>>> Is an open connected subspace of a homogeneous space homogeneous?
>
>> Surely not - I mean there's simply no reason why one would
>> imagine this is so.
>
>Because then one would have a base of open connected homogeneous
>sets any for locally connected and locally homogeneous space.
>
>> any open connected subset of R^n _is_ homogeneous.
>
>For R this is immediate, because an open connected subset is
>open interval, hence homeomorphic to R, which is homogeneous.
>
>> (Pf: If O is such a set and a, b in O then there exists a set B
>> homeomorphic to a closed ball containing a and b in the interior
>> and contained in O. Any two points of a ball are swapped by some
>> homeomorphism of the ball fixing points of the boundary - that
>> homeomorphism hence extends to a homeomorphism of O.)
>
>Ok, I suppose. Seems alright.
>
>Aren't open connected subsets of R^n homeomorphic to an open ball?
>No, they have to be simply connected. Then that is so.

Not for n > 2. A spherical shell in R^3 is simply connected but
not homeomorphic to a ball.

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

David C. Ullrich

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