## Hausdorff convergence and frontier convergence

1) Prove giving a counterexample that the Hausdorff convergence of a sequence of open sets does not imply the Hausdorff convergence of the boundary of to the boundary of .

2) Let be a sequence of compact sets converging in the Hausdorff distance to . Prove that the set has at least a point of accumulation and every such point satisfies .

Before trying the to solve this problem, the following theorems could be helpful:

a) If are compacts then , where . Therefore is equivalent to converges uniformly to .

b) If is a sequence of compacts included in a fixed (great) compact then it exists some compact in such that there exists a sub-sequence which converges to in the Hausdorff distance.

(Hint: If you can include all compacts in 2) in a bigger compact , then their boundaries will also be a sequence of compacts in , and therefore their sequence will have at least one accumulation point.)

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!

Multumesc.