On the topology of free paratopological groups

The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group F(X) on a Tychonoff space X. In this paper, an analogue of Joiner's lemma for the free paratopological group FP (X) on a T-1 space X is proved. Using this, it is shown that the following conditions are equivalent for a space X: (1) X is T-1; (2) FP (X) is T-1; (3) the subspace X of FP (X) is closed; (4) the subspace X-1 of FP (X) is discrete; (5) the subspace X-1 is T-1; (6) the subspace X-1 is closed ; and (7) the subspace FP (n)(X) is closed for all n is an element of N, where FP (n)(X) denotes the subspace of FP (X) consisting of all words of length at most n.