Publications de l’Institut Mathématique, Nouvelle Série Vol. 101[115], pp. 213–221 (2017) 

THE MAPPING ${i}_{2}$ ON THE FREE PARATOPOLOGICAL GROUPSFucai Lin, Chuan LiuSchool of mathematics and statistics, Minnan Normal University, Zhangzhou, P. R. China; Department of Mathematics, Ohio University Zanesville Campus, Zanesville, USAAbstract: 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 quasiuniformity 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 quasiuniformity Classification (MSC2000): 22A30; 54D10; 54E99; 54H99 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 24 Apr 2017. This page was last modified: 11 May 2017.
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