On topologization of countably generated algebras

W. Żelazko

Studia Mathematica (1994)

  • Volume: 112, Issue: 1, page 83-88
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Żelazko, W.. "On topologization of countably generated algebras." Studia Mathematica 112.1 (1994): 83-88. <http://eudml.org/doc/216139>.

@article{Żelazko1994,
abstract = {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.},
author = {Żelazko, W.},
journal = {Studia Mathematica},
keywords = {example of a semitopological algebra with every commutative subalgebra topological; complex countably generated algebra; topological algebra; continuum hypothesis},
language = {eng},
number = {1},
pages = {83-88},
title = {On topologization of countably generated algebras},
url = {http://eudml.org/doc/216139},
volume = {112},
year = {1994},
}

TY - JOUR
AU - Żelazko, W.
TI - On topologization of countably generated algebras
JO - Studia Mathematica
PY - 1994
VL - 112
IS - 1
SP - 83
EP - 88
AB - 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.
LA - eng
KW - example of a semitopological algebra with every commutative subalgebra topological; complex countably generated algebra; topological algebra; continuum hypothesis
UR - http://eudml.org/doc/216139
ER -

References

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  1. [1] A. Mallios, Topological Algebras. Selected Topics, North-Holland, Amsterdam, 1986. Zbl0597.46046
  2. [2] V. Müller, On topologizable algebras, Studia Math. 99 (1991), 149-153. Zbl0764.46046
  3. [3] H. H. Schaefer, Topological Vector Spaces, Springer, New York, 1971. 
  4. [4] W. Żelazko, Selected Topics in Topological Algebras, Aarhus Univ. Lecture Notes No. 31, 1971. Zbl0221.46041
  5. [5] W. Żelazko, On certain open problems in topological algebras, Rend. Sem. Mat. Fis. Milano 59 (1989) (1992), 49-58. Zbl0755.46019
  6. [6] W. Żelazko, Example of an algebra which is non-topologizable as a locally convex algebra, Proc. Amer. Math. Soc. 110 (1990), 947-949. Zbl0727.46025

Citations in EuDML Documents

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  1. A. Kokk, W. Żelazko, On vector spaces and algebras with maximal locally pseudoconvex topologies
  2. M. Wojciechowski, W. Żelazko, Non-uniqueness of topology for algebras of polynomials
  3. R. Frankiewicz, G. Plebanek, An example of a non-topologizable algebra
  4. W. Żelazko, Concerning topologization of real or complex algebras
  5. Mati Abel, Topological algebras with maximal regular ideals closed
  6. W. Żelazko, A characterization of commutative Fréchet algebras with all ideals closed

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