A Borel linear subspace of $\mathbb R^\omega$ that cannot be covered by countably many closed Haar-meager sets

TL;DR

The paper constructs a specific Borel linear subspace of an infinite-dimensional product space that defies coverage by countably many closed Haar-meager sets, shedding light on the structure of large sets in topological vector spaces.

Contribution

It introduces a novel example of a Borel linear subspace in an infinite product space that cannot be covered by countably many closed Haar-meager sets, advancing understanding of large set classes.

Findings

Existence of a non-coverable Borel linear subspace in $ ext{R}^ ext{omega}$

Insights into the relationship between large sets and Kuczma--Ger classes

Applications to topological vector space theory

Abstract

We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^\omega$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma--Ger classes in the topological vector spaces $\mathbb R^n$ for $n\le \omega$.