On the topology of free paratopological groups. Ⅱ

摘要

Let FP(X) be the free paratopological group on a topological space X. For n e N, denote by FP_n(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 FP_n(X). In this paper a neighbourhood base at the identity e in FP_2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i2 : (X◎ X~(-1)d ◎{e})~2 → FP_2(X) is a quotient mapping, where X is a T1 space and X~(-1)d denotes the set X~(-1) equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T_1 space. Several specific spaces and classes of spaces are also examined. For example,i_2 is a quotient mapping for every countable subspace of R, i_2 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.