I am confused by the above statement with the follwing version of Baire's category theorem:
If a non-empty complete metric space $(M,d)$ is the countable union of closed sets, then one of these closed sets has non-empty interior.
A singleton in a complete metric space is a complete metric space. By Baire's theorem the singleton should have an interior point, which I believe it does not.
Best Answer
Yes, it has an interior point. If $p\in M$, then $p$ is an interior point of $\{p\}$, if we see $\{p\}$ as a metric subspace of $M$. Of course, in $M$, the set $\{p\}$ has, in general, no interior point (it has only if $p$ is an isolated point of $M$).