Is $X$ is second countable implies or equivalent to Every closed(or open) set in $X$ is a countable intersection of open sets in $X$?
I know, metric space implies this by Closed set as a countable intersection of open sets, and Every open set in $\mathbb{R}$ is a countable union of closed sets. Also I know metric space is 1st countable but not 2nd countable.
If so How one can prove this statement?
Best Answer
The natural numbers $\ge 2$ endowed with the topology of the subsets that contain the divisors of each of their elements is second-countable, but not $G_\delta$.