Résume | For any point in a Polish space and any subset it is not a part of, we define the type of this
point for this subset as an integer between 1 and 3 depending on notions of compactness. This
allows us to formulate a homeomorphism extension lemma on zero-dimensional Polish spaces.
If X and A are two such spaces and f is a homeomorphism between X' and A' that preserves
the type of points for X\X' and A\A' respectively, f can be extended to X and A if and only if
X\X' and A\A' are of same cardinal and their closure have the same compactness. This lemma
can be used to prove that the set of zero-dimensional Polish spaces that are homeormorphic to
a non-locally compact countable Polish space X is co-analytic in F(|N^|N), for any such X. |