In metric spaces, every open set is a countable union of closed sets.
is the converse true?
A topological space with the property "every open set is a countable union of closed sets" has to be metrizable?
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Sign up to join this communityIn metric spaces, every open set is a countable union of closed sets.
is the converse true?
A topological space with the property "every open set is a countable union of closed sets" has to be metrizable?
No the Sorgenfrey line (a.k.a. $\mathbb{R}$ in the lower limit topology) is perfectly normal (as this property is called) but not metrisable.
A compact counterexample is the Double Arrow space, which is a related example.
No. Take the indiscrete topology as a counterexample.
If that were true, then any countable T$_1$-space would be metrizable. In fact there are lots of countable Hausdorff spaces that aren't even first countable. For example, topologize $\mathbb N$ so that $$S\text{ is closed }\iff1\in S\text{ or }\sum_{n\in S}\frac1n\lt\infty.$$