On topologization of countably generated algebras

• Autorzy: W. Żelazko
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46H05: General theory of topological algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994-1995
• Tom: 112
• Numer: 1
• Strony: 83-88

Daty

wydano

1994

otrzymano

1994-03-23

poprawiono

1994-04-27

Twórcy

autor

W. Żelazko

• Institute of Mathematics, Polish Academy of Sciences, P.O. Box 137, 00-950 Warszawa, Poland

Bibliografia

• [1] A. Mallios, Topological Algebras. Selected Topics, North-Holland, Amsterdam, 1986.
• [2] V. Müller, On topologizable algebras, Studia Math. 99 (1991), 149-153.
• [3] H. H. Schaefer, Topological Vector Spaces, Springer, New York, 1971.
• [4] W. Żelazko, Selected Topics in Topological Algebras, Aarhus Univ. Lecture Notes No. 31, 1971.
• [5] W. Żelazko, On certain open problems in topological algebras, Rend. Sem. Mat. Fis. Milano 59 (1989) (1992), 49-58.
• [6] W. Żelazko, Example of an algebra which is non-topologizable as a locally convex algebra, Proc. Amer. Math. Soc. 110 (1990), 947-949.