On the topology of free paratopological groups
Ali Sayed Elfard, Peter Nickolas

TL;DR
This paper extends Joiner's lemma to free paratopological groups on T_1 spaces, establishing key topological equivalences and properties that deepen understanding of their structure.
Contribution
It proves an analogue of Joiner's lemma for free paratopological groups and characterizes when these groups are T_1 based on subspace properties.
Findings
Equivalence of T_1 conditions for X and its free paratopological group
Characterization of subspace closure and discreteness in FP(X)
Conditions under which subspaces of FP(X) are closed or discrete
Abstract
The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group on a Tychonoff space. In this paper, an analogue of Joiner's lemma for the free paratopological group on a space is proved. Using this, it is shown that the following conditions are equivalent for a space : (1) is ; (2) is ; (3) the subspace of is closed; (4) the subspace of is discrete; (5) the subspace is ; (6) the subspace is closed; and (7) the subspace is closed for all , where denotes the subspace of consisting of all words of length at most .
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