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Topic: homeomorphisms of compact sets and the hausdorff distance
Replies: 7   Last Post: Nov 9, 2010 11:52 PM

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Rodolfo Conde

Posts: 16
Registered: 3/24/10
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...


> 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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