# Precompact dots

Let $M, N$ be metric spaces such that $M$ is connected and $N$ is complete and locally compact. Let $\mathcal{F}$ be a uniformly equicontinuous family of functions from $M$ to $N$ such that there exists $x_0\in M$ with the property $(\star)$ that $\mathcal{F}(x_0)$ is precompact. Prove that all points $x\in M$ have property $(\star)$.

# No point left behind

Every infinite set admits a metric with no isolated points.

# Compact function spaces

Let $K$ be a compact metric space, $\varphi:K\to(0,\infty)$ a continuous function and $A=\{f\in C(K) : |f(x)|\leq\varphi(x)\}$. Prove that $A$ is compact iff $K$ is finite.

# 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 $n$.

# Compact isometries

As seen inĀ this post, any isometry from a compact metric space $K$ to itself is surjective, so the set $I(K)$ of all such isometries is actually a group.

Give $I(K)$ a reasonable metric and prove that it is compact as well. What can one say about the sequence of iterates $K, I(K), I(I(K)), \dots$?

# Infinitely far, yet very close

How much does the theory of metric spaces change if we let the distance function take the value $+\infty$? Some natural examples of “extended metric spaces”:

• The extended real numbers.
• Any function space $Y^X$ where $Y$ is a metric space (or an extended metric space) with the usual supremumĀ distance: $d(f,g) = \sup_{x\in X} d(f(x),g(x))$.