Say we have two compact Hausdorff spaces $X,Y$ with open subsets $A \subset X, B \subset Y$ such that both are homeomorphic to the open unit disc and their boundaries $\partial A,\partial B$ are homeomorphic to the unit circle. If there is some given homeomorphism $h: \partial A \to \partial B$, can this be extended to a homeomorphism $\bar{h}: \bar{A} \to \bar{B}$?
This fact seems intuitively clear but I can't seem to find a good argument.
Best Answer
Let A and B be open discs. Let X be the usual compactification of A (i.e. closed disc) but construct Y by taking the usual compactification of B and identifying opposite points of the boundary. Both remainders are circles but X and Y are not homeomorphic.