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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.

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