Let . Prove that the group of homeomorphisms of the -dimensional ball that fix its boundary contains a copy of the free group on generators.
Construct a path-connected metric space and a discontinuous function such that is continuous for any continuous path .
Find a compact metric space that does not embed in for any .
Construct a countable, connected topological space with at least two points, satisfying the highest separability axiom you can.
Notice that if two points can be separated by a continuous function, then a connected space with more than one point is uncountable, so your space cannot be (or higher).
Construct two homeomorphic closed subsets of some Euclidean space such that their complements are not homeomorphic, but show that if are closed homeomorphic subsets of such that lies in and lies in , their complements in are homeomorphic.
Consider an isometry from a compact metric space to itself, this is simply a function that preserves distances. Notice that this implies that the function is continuous and moreover that it is injective. What we have to prove is that such a function is also surjective. Observe that this implies that the function is indeed a homeomorphism.
A consequence of this fact is that the category that has compact metric spaces as objects and isometries as arrows enjoys a Cantor-Schröder-Bernstein-like property.