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Topological Groups and Related Structures
- Publisher
- Wiley
- Copyright
- © London Mathematical Society
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/blms/bds031
- Publisher site
- See Article on Publisher Site
Abstract
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 T1 space X is proved. Using this, it is shown that the following conditions are equivalent for a space X: (1) X is T1; (2) FP (X) is T1; (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 T1; (6) the subspace X−1 is closed and (7) the subspace FP n(X) is closed for all n∈ℕ, where FP n(X) denotes the subspace of FP (X) consisting of all words of length at most n.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: Dec 1, 2012