Countable intersection of nested non-empty closed sets in ZF

Question

For all sequences of nested (non-increasing with respect to set-inclusion) non-empty closed sets, their intersection is non-empty.

What are the minimum conditions on a topological space for it to satisfy this property in ZF (without any choice)?

Answer 1

This true iff every countable open cover has a finite subcover. The proof involves just taking complements and noting that if $U_n$ is any sequence of open sets then $U_1 \cup U_2\cup...\cup U_n$ is an increasing sequence of open sets whose union is the same as the union of the $U_i$'s.