Let be metric spaces such that is connected and is complete and locally compact. Let be a uniformly equicontinuous family of functions from to such that there exists with the property that is precompact. Prove that all points have property .
Every infinite set admits a metric with no isolated points.
Let be a compact metric space, a continuous function and . Prove that is compact iff is finite.
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 .
As seen in 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 ?
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 supremum distance: .