Recall that a space $X$ is Menger if for each sequence $(\mathcal{U}_n)_{n\in\omega}$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)_{n\in\omega}$ such that, for each $n\in \omega$, $\mathcal{V}_n$ is a finite subcollection of $\mathcal{U}_n$ and $\bigcup\{\mathcal{V}_n:n\in\omega\}$ is an open cover of $X$.

On the other hand, given a space $X$, the Alexandroff duplicate $A(X)$ is defined as follows: The underlying set is $X\times\{0,1\}$, each point of $X\times\{1\}$ is isolated and a basic open neighbourhood for a point $\langle x, 0 \rangle \in X\times\{0\}$ is a set of the form $U\times\{0, 1\}\setminus\{\langle x, 1\rangle\}$ where $U$ is an open neighbourhood of $x$ in $X$.

It turns out that for several topological properties $\mathcal{P}$, it occurs that a space $X$ has the property $\mathcal{P}$ if and only if its Alexandroff duplicate $A(X)$ has the property $\mathcal{P}$. Examples of such properties are compactness, Lindelofness, normality, countably paracompactness, etc. I have found in some papers that, for the Menger property is well-known that this fact also occurs; that is, a space $X$ is Menger if and only if its Alexandroff duplicate $A(X)$ is Menger. The problem is that I have not been able to find a reference where this fact has been showed for the first time. Does anyone knows any reference to be cited where this result had been showed? I agree that showing this fact is not dificult.

Thanks for any help!