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 | | From: | Valeriu Anisiu | | Subject: | Re: Homeogeneous Spaces | | Date: | Sun, 23 Jan 2005 23:53:12 +0000 (UTC) |
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 | On 23 Jan 2005, William Elliot 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?
>
>Counter examples, of course, are welcome.
>
>----
No.
S=[0,1] with the natural topology is homogeneous,
[0,1) is open and connected _in S_ but it is not homogeneous.
V. Anisiu
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| | From: | Ron Sperber | | Subject: | Re: Homeogeneous Spaces | | Date: | Sun, 23 Jan 2005 22:58:26 -0500 |
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| Valeriu Anisiu wrote:
> On 23 Jan 2005, William Elliot 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?
>>
>>Counter examples, of course, are welcome.
>>
>>----
>
>
> No.
> S=[0,1] with the natural topology is homogeneous,
> [0,1) is open and connected _in S_ but it is not homogeneous.
>
> V. Anisiu
>
[0,1] is NOT homogeneous. There is no homeomorphism f:[0,1]->[0,1] with
f(0)=1/2 for example.
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