On the topology of free paratopological groups. II

Let FP ( X ) be the free paratopological group on a topological space X. For n ∈ N , denote by FP n ( X ) the subset of FP ( X ) consisting of all words of reduced length at most n, and by i n 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 i 2 : ( X ⊕ X d − 1 ⊕ { e } ) 2 → FP 2 ( X ) is a quotient mapping, where X is a T 1 space and X d − 1 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 i 2 is a quotient mapping.