- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Here's this week's problem.
-----
Problem: A space $X$ is locally metrizable if each point $x\in X$ has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space $X$ is metrizable if and only if it is locally metrizable.
-----
Hint:
Remember to read the POTW submission guidelines to find out how to submit your answers!
-----
Problem: A space $X$ is locally metrizable if each point $x\in X$ has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space $X$ is metrizable if and only if it is locally metrizable.
-----
Hint:
Show that $X$ is a finite union of open subspaces, each of which has a countable basis.
Remember to read the POTW submission guidelines to find out how to submit your answers!