Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.
@article{bwmeta1.element.bwnjournal-article-cmv77z1p1bwm,
author = {Vladimir Pestov and Dmitri Shakhmatov},
title = {A relatively free topological group that is not varietal free},
journal = {Colloquium Mathematicae},
volume = {78},
year = {1998},
pages = {1-8},
zbl = {0901.22002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv77z1p1bwm}
}
Pestov, Vladimir; Shakhmatov, Dmitri. A relatively free topological group that is not varietal free. Colloquium Mathematicae, Tome 78 (1998) pp. 1-8. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv77z1p1bwm/
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