We prove that any real or complex countably generated algebra has a complete locally convex topology making it a topological algebra. Assuming the continuum hypothesis, it is the best possible result expressed in terms of the cardinality of a set of generators. This result is a corollary to a theorem stating that a free algebra provided with the maximal locally convex topology is a topological algebra if and only if the number of variables is at most countable. As a byproduct we obtain an example of a semitopological (non-topological) algebra with every commutative subalgebra topological.
@article{bwmeta1.element.bwnjournal-article-smv112i1p83bwm,
author = {W. \.Zelazko},
title = {On topologization of countably generated algebras},
journal = {Studia Mathematica},
volume = {108},
year = {1994},
pages = {83-88},
zbl = {0832.46042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv112i1p83bwm}
}
Żelazko, W. On topologization of countably generated algebras. Studia Mathematica, Tome 108 (1994) pp. 83-88. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv112i1p83bwm/
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