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Re: homeomorphisms of compact sets and the hausdorff distance
Posted:
Nov 8, 2010 10:43 PM
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El 08/11/2010 01:32 p.m., G. A. Edgar escribi?:
> In article<ib9e99$prv$1@login.math.ohio-state.edu>, Rodolfo Conde
> <rcm@gmx.co.uk> wrote:
>
>> Thanks everyone for your answers, now i got a refinement of my question:
>>
>> Take X=a TOP manifold, A=a compact TOP submanifold of X and suppose
>> that we have a sequence B_n of compact subsets of X such that each B_n
>> is a union of topological (open) balls in X,
>
Sorry, indeed this statement is weird. It should say "topological
(closed) balls in X", not open balls. Manifolds are, as usual,
metrizable and connected.
Could A and B_n have the same homotopy type with the given hypothesis ??
Thanks everyone...
Greetings...
> A bit strange statement. So, B_n is open? And compact? Of course
> this could happen if X is disconnected, so verify that it is what you
> want...
>
>> such that for any e> 0,
>> H(A, B_n)< e for n sufficiently far out in the secuence.
>>
>> It is not hard to see that it is not always true that for n large
>> enough, B_n is homeomorphic to A (X=q-euclidean space, A=m-submanifold,
>> and the B_n are unions of open q-balls in X with q>n).
>>
>> But, Can it be proven that for n large enough, A and B_n have the same
>> homotopy type ? examining some examples, this would seem plausible. It
>> would even seem that B_n can be some kind of "special neighborhood" for A
>>
>> Thanks in advance...
>>
>> Cheers...
>>
>
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