Linear homeomorphisms of function spaces and the position of a space in its compactification

TL;DR

This paper proves that the Menger property of a Tychonoff space is preserved under linear homeomorphisms of its function space, using invariants related to the space's position in its compactification.

Contribution

It introduces a new method to study invariants of linear homeomorphisms of function spaces by examining the space's placement in its compactification.

Findings

Menger property is preserved under linear homeomorphisms of $C_p(X)$.

Develops a novel approach linking space invariants to compactification positioning.

Provides an affirmative answer to a longstanding question for linear homeomorphisms.

Abstract

An old question of A.V. Arhangel'skii asks if the Menger property of a Tychonoff space $X$ is preserved by homeomorphisms of its function space $C_p(X)$. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of fuction spaces $C_p(X)$ by looking at the way $X$ is positioned in its (\v{C}ech-Stone) compactification.