# Free subgroups of the group of homeomorphisms

Let $n\geq 2$. Prove that the group of homeomorphisms of the $n$-dimensional ball that fix its boundary contains a copy of the free group on $2$ generators.

# Spooky geometry III

Construct a path-connected metric space $X$ and a discontinuous function $f:X\to \Bbb{R}$ such that $f\circ \sigma$ is continuous for any continuous path $\sigma:[0,1]\to X$.

# Spooky geometry II

Find a compact metric space that does not embed in $\Bbb{R}^n$ for any $n$.

# Connect the dots

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 $T_{3 \frac 12}$ (or higher).

# Complements of closed sets

Construct  two homeomorphic closed subsets $A,B$ of some Euclidean space such that their complements are not homeomorphic, but show that if $A,B$ are closed homeomorphic subsets of $\Bbb R^{n+m}$ such that $A$ lies in $\Bbb R^n\times 0$ and $B$ lies in $0\times\Bbb R^m$, their complements in $\Bbb R^{n+m}$ are homeomorphic.

# Self isometries of a compact metric space

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.