On the topology of free paratopological groups. II

Abstract

Let FP(X) be the free paratopological group on a topological space X. For n∈N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and by in the natural mapping from (X⊕X−1⊕{e})n to FPn(X). In this paper a neighbourhood base at the identity e in FP2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i2:(X⊕Xd−1⊕{e})2→FP2(X) is a quotient mapping, where X is a T1 space and Xd−1 denotes the set X−1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T1 space. Several specific spaces and classes of spaces are also examined. For example, i2 is a quotient mapping for every countable subspace of R, i2 is not a quotient mapping for any uncountable compact subspace of R, and it is undecidable in ZFC whether an uncountable subspace of R exists for which i2 is a quotient mapping.