The $k$-spaces property of free Abelian topological groups over non-metrizable La\v{s}nev spaces

TL;DR

This paper investigates the $k$-space property of free Abelian topological groups over non-metrizable Lašnev spaces, establishing conditions under which subspaces of bounded word length reflect the $k$-space property of the entire group.

Contribution

It provides a characterization of when certain subspaces of free Abelian topological groups over Lašnev spaces are $k$-spaces, extending previous results and exploring the influence of set-theoretic assumptions.

Findings

$A_4(X)$ is a $k$-space iff $A(X)$ is a $k$-space for non-metrizable Lašnev spaces.

Under $lat= ext{ω}_1$, $A_3(X)$ is a $k$-space iff $A(X)$ is a $k$-space.

Under $lat> ext{ω}_1$, there exists a Lašnev space where $A_3(X)$ is a $k$-space but $A(X)$ is not.

Abstract

Given a Tychonoff space $X$, let $A(X)$ be the free Abelian topological group over $X$ in the sense of Markov. For every $n\in\mathbb{N}$, let $A_n(X)$ denote the subspace of $A(X)$ that consists of words of reduced length at most $n$ with respect to the free basis $X$. In this paper, we show that $A_4(X)$ is a $k$-space if and only if $A(X)$ is a $k$-space for the non-metrizable La\v{s}nev space $X$, which gives a complementary for one result of K. Yamada's. In addition, we also show that, under the assumption of $\flat=\omega_1$, the subspace $A_3(X)$ is a $k$-space if and only if $A(X)$ is a $k$-space for the non-metrizable La\v{s}nev space $X$. However, under the assumption of $\flat>\omega_1$, we provide a non-metrizable La\v{s}nev space $X$ such that $A_3(X)$ is a $k$-space but $A(X)$ is not a $k$-space.