# Does there exist a complete metric space which is Rothberger (or Menger) but not Hurewicz?

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

1. Menger 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 $$\cup_{n\in\mathbb{N}}\mathcal{V}_n$$ is an open cover of $$X$$.
2. 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 $$n$$.
3. 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 (or Menger) but not Hurewicz.

• I have already posted this question on MSE but did not get answer. math.stackexchange.com/questions/3817583/… math.stackexchange.com/questions/3817530/… – Nur Alam Sep 9 at 17:41
• If I remember well, a complete metric space is Menger iff it is $\sigma$-compact (shown by Hurewicz himself). Hence a complete metric Menger space is Hurewicz. – Mathieu Baillif Sep 11 at 8:25
• @Mathieu This should be the paper by Hurewicz in which the implication you mentioned is proved, but my German is a bit rusty. In any case more recently the result was extended to all Cech-complete spaces by Tall and Tokgöz – Alessandro Codenotti Sep 11 at 16:00
• If a metric space is a Rothberger space, then it is separable, and it contains no subspace homeomorphic to the Cantor set. Hence a complete metric space is a Rothberger space if and only if it is countable and therefore a Hurewicz space. (I posted this as an answer which I deleted it because I saw it was already answered in comments.} – bof Sep 25 at 4:26
• Definition 2.Hurewicz space, "but finitely many 𝑛" -- is there a typo? – Wlod AA Sep 25 at 7:01