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

TL;DR

This paper explores the existence of a Tychonoff space whose function space's remainder in some compactification is Menger but not sigma-compact, linking this to the existence of a Menger ultrafilter.

Contribution

It demonstrates the consistency of such a space's existence, connecting topological properties with ultrafilter theory.

Findings

Existence of a Tychonoff space with Menger remainder in some compactification is consistent.

Such a space's existence follows from the existence of a Menger ultrafilter.

The result addresses a question posed by Bella, Tokg"os, and Zdomskyy.

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.