Mapping i2 on the free paratopological groups

Abstract

Let FP(X) be the free paratopological group over a topological space X. For each nonnegative integer n ? N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and in by the natural mapping from (X ? X?1 ? {e})n to FPn(X). We prove that the natural mapping i2:(X ? X?1 d ?{e})2 ? FP2(X) is a closed mapping if and only if every neighborhood U of the diagonal ?1 in Xd x X is a member of the finest quasi-uniformity on X, where X is a T1-space and Xd denotes X when equipped with the discrete topology in place of its given topology.