Does second countablity implies closed (or open) set in $X$ is a countable intersection of open sets in $X$?

Question

Is $X$ is second countable implies or equivalent to Every closed(or open) set in $X$ is a countable intersection of open sets in $X$?

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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?

Answer 1

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 $G_\delta$.

Answer 2

The Double Arrow space ($[0,1] \times \{0,1\}$ in the order topology induced from the lexicographer order) is not second countable but is compact, connected and every closed subset is a countable intersection of open sets (it’s perfectly normal). So it’s not equivalent.