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Free topological groups of spaces and their subspaces

✍ Scribed by Ol'ga V. Sipacheva

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
266 KB
Volume
101
Category
Article
ISSN
0166-8641

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✦ Synopsis

We prove that if X is a Tychonoff topological space, Y a subspace of X, and every bounded continuous pseudometric on Y can be extended to a continuous pseudometric on X, then the free topological group F M (Y ) coincides with the topological subgroup of F M (X) generated by Y . For this purpose, a new description for the topology of a free topological group in terms of continuous pseudometrics and group seminorms is given. It follows from what has been shown by UspenskiΘ‹ that this result implies the Weil completeness of F M (X) for any DieudonnΓ© complete X. It is also proved that if dim X = 0, then ind F M (X) = 0.

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