Every infinite set admits a metric with no isolated points.

# metric spaces

# Compact function spaces

Let be a compact metric space, a continuous function and . Prove that is compact iff is finite.

# Spooky geometry III

Construct a path-connected metric space and a discontinuous function such that is continuous for any continuous path .

# Spooky geometry II

Find a compact metric space that does not embed in for any .

# Compact isometries

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

Give a reasonable metric and prove that it is compact as well. What can one say about the sequence of iterates ?

# Infinitely far, yet very close

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

- The extended real numbers.
- Any function space where is a metric space (or an extended metric space) with the usual supremumĀ distance: .

# No swapping

There is no continuous function such that and .