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 afinitary Rothberger propertyif 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.