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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.

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    $\begingroup$ I have already posted this question on MSE but did not get answer. math.stackexchange.com/questions/3817583/… math.stackexchange.com/questions/3817530/… $\endgroup$ – Nur Alam Sep 9 at 17:41
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    $\begingroup$ 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. $\endgroup$ – Mathieu Baillif Sep 11 at 8:25
  • $\begingroup$ @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 $\endgroup$ – Alessandro Codenotti Sep 11 at 16:00
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    $\begingroup$ 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.} $\endgroup$ – bof Sep 25 at 4:26
  • $\begingroup$ Definition 2.Hurewicz space, "but finitely many 𝑛" -- is there a typo? $\endgroup$ – Wlod AA Sep 25 at 7:01

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