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