Home > Problem Solving, Topology > Hausdorff convergence and frontier convergence

Hausdorff convergence and frontier convergence


1) Prove giving a counterexample that the Hausdorff convergence of a sequence of open sets \Omega_n does not imply the Hausdorff convergence of the boundary of \Omega_n to the boundary of \Omega.
2) Let K_n be a sequence of compact sets converging in the Hausdorff distance to K. Prove that the set \partial K_n has at least a point of accumulation and every such point L satisfies \partial K \subset L \subset K.

Before trying the to solve this problem, the following theorems could be helpful:
a) If K_1,K_2 are compacts then d^H(K_1,K_2)= \|d_{K_1}-d_{K_2}\|_{L^\infty}, where d_K=d(\cdot,K). Therefore K_n \xrightarrow[]{H} K is equivalent to d_{K_n}-d_K converges uniformly to 0.

b) If K_n is a sequence of compacts included in a fixed (great) compact B then it exists some compact K in B such that there exists a sub-sequence (K_{n_k}) \subset (K_n) which converges to K in the Hausdorff distance.
(Hint: If you can include all compacts in 2) in a bigger compact B, then their boundaries will also be a sequence of compacts in B, and therefore their sequence will have at least one accumulation point.)

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  1. March 21, 2011 at 5:34 pm

    Hello, pare foarte interesant; chestiile astea tin de fapt mai mult de probleme topologice de frontiera. La unibuc nu avem asa ceva, dar am fost la o bursa in Spania si acolo am intalnit un prof Buttazzo care era pasionat de partea asta de a “repara” contururi. Succes!

  2. March 21, 2011 at 7:57 pm

    Multumesc.

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