The mapping $i_2$ on the free paratopological groups

Abstract

Let $FP(X)$ be the free paratopological group over a topological space $X$. For each non-negative integer $n\in \mathbb{N}$, denote by $F{P}_n(X)$ the subset of $FP(X)$ consisting of all words of reduced length at most $n$, and $i_n$ by the natural mapping from $(X⨁X^{-1}⨁\{e\}{)}^n$ to $F{P}_n(X)$. In this paper, we mainly improve some results of A.S. Elfard and P. Nickolas's [On the topology of free paratopological groups. II, Topology Appl., 160(2013), 220--229.]. The main result is that the natural mapping $i_2:(X⨁X_d^{-1}⨁\{e\}{)}^2⟶F{P}_2(X)$ is a closed mapping if and only if every neighborhood $U$ of the diagonal ${\Delta }_1$ in $X_d\times X$ is a member of the finest quasi-uniformity on $X$, where $X$ is a $T_1$-space and $X_d$ denotes $X$ when equipped with the discrete topology in place of its given topology.

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References in corpus (1)

• Free paratopological groups