A non-discrete space $X$ with $C_p(X)$ Menger at infinity

Abstract

In a paper by Bella, Tokg\"os and Zdomskyy it is asked whether there exists a Tychonoff space $X$ such that the remainder of $C_p(X)$ in some compactification is Menger but not $\sigma$-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.