
Re: homeomorphisms of compact sets and the hausdorff distance
Posted:
Nov 8, 2010 10:43 PM


El 08/11/2010 01:32 p.m., G. A. Edgar escribi?: > In article<ib9e99$prv$1@login.math.ohiostate.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=qeuclidean space, A=msubmanifold, >> and the B_n are unions of open qballs 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... >> >

