
Re: homeomorphisms of compact sets and the hausdorff distance
Posted:
Nov 9, 2010 11:52 PM


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...
Cheers...

