What is the source of the claim that "Strongly zero-dimensional and zero-dimensional is equivalent for separable metrizable spaces"? My university lecture told me that this is true, but I fail to find it anywhere in literature. Could you provide a source for this?
Definitions
A Hausdorff topological space $X$ is zero dimensional if for every point $x$ of $X$
and every neighborhood $U$ of $x$ in $X$, there exists a nonempty clopen subset $V$ of $X$ such that $x \in V \subset U$.
The clopen basis of any zero-dimensional space is a collection of clopen sets that is closed under complements and finite intersections.
A Hausdorff topological space $X$ is said to be strongly zero-dimensional whenever for every closed subset $A$ of $X$ and every open subset $U$ of $X$ such that $A \subseteq U$, there exists a clopen subset $V$ of $X$ such that $A \subseteq V \subseteq U.$
Best Answer
What you call "strongly zero-dimensional" is usually called ultranormal. In this common notation a Tychonoff space is called strongly zero-dimensional, if its Stone-Cech compactification is zero-dimensional. A space is ultranormal, iff it is normal and strongly zero-dimensional. In particular, for metric spaces, ultranormality is the same as strong zero-dimensionality. See, for instance here.
In the paper cited in the above link, you also find a proof for "Every Lindelof zero-dimensional space is ultraparacompact" (hence ultranormal), thus answering your question, since a separable metric space is, of course, Lindelof. See also Engelking, General topology 6.2.7.