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The $k$-spaces property of free Abelian topological groups over non-metrizable Lašnev spaces

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 Lašnev space $X$, which gives a complementary for one result of K. Yamada's. In addition, we also show that, under the assumption of $\flat=ω_1$, the subspace $A_3(X)$ is a $k$-space if and only if $A(X)$ is a $k$-space for the non-metrizable Lašnev space $X$. However, under the assumption of $\flat>ω_1$, we provide a non-metrizable Lašnev space $X$ such that $A_3(X)$ is a $k$-space but $A(X)$ is not a $k$-space.