# Does there exist a complete metric space which is Rothberger but not Hurewicz?

A topological space $$X$$ is said to be a

1. Hurewicz space if for each sequence $$(\mathcal{U}_n)$$ of open covers of $$X$$ there is a sequence $$(\mathcal{V}_n)$$ such that for each $$n$$ $$\mathcal{V}_n$$ is a finite subset of $$\mathcal{U}_n$$ and each $$x\in X$$ belongs to $$\cup\mathcal{V}_n$$ for all but finitely many $$n$$.
2. Rothberger space if for each sequence $$(\mathcal{U}_n)$$ of open covers of $$X$$ there is a sequence $$(U_n)$$ such that for each $$n$$ $$U_n\in\mathcal{U}_n$$ and $$\cup_{n\in\mathbb{N}}U_n=X$$.

I did not find an example of a complete metric space which is Rothberger but not Hurewicz.