# Rothberger property for finite covers

Let us recall that a topological space $$X$$ has the Rothberger property if for any sequence $$(\mathcal U_n)_{n\in\omega}$$ of open covers of $$X$$ there exists a sequence $$(U_n)_{n\in\omega}\in\prod_{n\in\omega}\mathcal U_n$$ such that $$X=\bigcup_{n\in\omega}U_n$$.

I am interested in the finitary version of the Rothberger property.

Definition. A topological space $$X$$ is defined to have a finitary Rothberger property if for any sequence $$(\mathcal U_n)_{n\in\omega}$$ of finite open covers of $$X$$ there exists a a sequence $$(U_n)_{n\in\omega}\in\prod_{n\in\omega}\mathcal U_n$$ such that $$X=\bigcup_{n\in\omega}U_n$$.

It is clear that each topological space $$X$$ with the Rothberger property has the finitary Rothberger property.

On the other hand, each subspace $$X\subset \mathbb R$$ with the finitary Rothberger property has strong measure zero and hence is countable if the Borel conjecture is true.

Question 1. Is there a (necessarily consistent) example of a topological space that has the finitary Rothberger property but fails to have the Rothberger property?

By a result of Fremlin and Miller, a metrizable space $$X$$ has the Rothberger property if and only if $$X$$ has strong measure zero with respect to any metric $$d$$ generating the topology of $$X$$. The latter means that for any sequence of positive real numbers $$(\varepsilon_n)_{n\in\omega}$$ there exists a cover $$\{U_n\}_{n\in\omega}$$ of $$X$$ such that each set $$U_n$$ has diameter $$<\varepsilon_n$$ with respect to the metric $$d$$.

By analogy it can be shown that a metrizable space $$X$$ has the finitary Rothberger propery if and only if $$X$$ has strong measure zero with respect to any totally bounded metric generating the topology of $$X$$.

Added in Edit. After the answer of Boaz Tsaban I looked at the Handbook's article Special subsets of the real line" by Arnold Miller and found that my question is equivalent to the question of Rothberger: Is every $$C'$$ set a $$C''$$ set, which was still open in 1984, but has been answered in negative by Fremlin and Miller in 1985 and eventually published in 1988.