THE MAPPING $i_2$ ON THE FREE PARATOPOLOGICAL GROUPS

Fucai Lin, Chuan Liu

Abstract: Let $FP\left(X\right)$ be the free paratopological group over a topological space $X$. For each nonnegative integer $n\in \mathbb{N}$, denote by $F{P}_n\left(X\right)$ the subset of $FP\left(X\right)$ consisting of all words of reduced length at most $n$, and $i_n$ by the natural mapping from ${(X\oplus X^{-1}\oplus \left\{e\right\})}^n$ to $F{P}_n\left(X\right)$. We prove that the natural mapping $i_2:{(X\oplus X_d^{-1}\oplus \left\{e\right\})}^2\to F{P}_2\left(X\right)$ 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.

Keywords: free paratopological groups; quotient mappings; closed mappings; finest quasi-uniformity

Classification (MSC2000): 22A30; 54D10; 54E99; 54H99

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