Is it always possible to extend a homeomorphism from a closed ball in the plane onto a closed disk in a surface to their neighbourhoods?

Question

Let $S$ be a two dimensional surface and $D$ a closed disk contained in $S$. Is it true that if $f$ is a homeomorphism from a closed ball $B$ in the plane onto $D$, then $f$ extends to a homeomorphism from a neighbourhood of $B$ in the plane onto a neighbourhood of $D$ in the surface?

Answer 1

Assuming that you don't mean manifold with boundary (in which case it's not true, since $D$ might already be the whole space!) then yes, it's true. One way to do it is with the Tubular Neighborhood Theorem by switching to the smooth category.

Let $J$ be the boundary curve of your disc in $S$. Then it has a tubular neighborhood, which is diffeomorphic to a neighborhood of $J$ in its normal tangent bundle. For a simple closed curve $J$ that's just an open annulus. This neighborhood is separated into two components by $J$, one of which will meet the interior of your disc. Keep the other one and glue your annulus homeomorphism and disc homeomorphism together along $J$.

If it's annoying having to go into the smooth category, just be aware that these sorts of theorems get VERY hard without appealing to smooth, algebraic/PL or complex-analytic techniques.