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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 9, 2010 11:52 PM
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El 09/11/2010 09:12 a.m., G. A. Edgar escribi?:
> Here is an example to think about. You can tell us whether it is
> within the conditions you intend ... All are subsets of the plane. B_n
> is the annulus, center at the origin, inner radius 1/n, outer radius 1.
> The limit A is the disk, center at the origin, radius 1. So in this
> example, the homotopy type of the limit A is different from the
> homotopy types of the approximations B_n .

Thanks. this example is good and shows that with the conditions i have
stated the proposition is false. I now want to add an additional
condition. I will state the problem in full (again, for clarity) with
the new condition:

Let X be a TOP manifold and M a compact TOP submanifold of X. Suppose
that we have a sequence of compact sets B_n such that for each n:
1.- B_n is a finite union of topological closed balls in X.
2.- M \subset B_n \subset N(2^{-n}, M)

where N(2^{-n}, M) = union_{x\in M} N(2^{-n}, x)

Also, suppose that for each e > 0, for n large enough, H(B_n, M) < e
(Hausdorff distance)

Question: Is it true for B_n sufficiently far out in the sequence, B_n
and M have the same homotopy type ?

Thanks in advance...


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