The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.

Taras Banakh; Anatolij Plichko

RACSAM (2006)

  • Volume: 100, Issue: 1-2, page 31-37
  • ISSN: 1578-7303

Abstract

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Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space X with meager hyperspace K(X). Also under (d = c) we construct a metrizable separable noncomplete linear metric space with hereditarily Baire hyperspace. We do not know if such a space can be constructed in ZFC.

How to cite

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Banakh, Taras, and Plichko, Anatolij. "The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.." RACSAM 100.1-2 (2006): 31-37. <http://eudml.org/doc/41640>.

@article{Banakh2006,
abstract = {Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space X with meager hyperspace K(X). Also under (d = c) we construct a metrizable separable noncomplete linear metric space with hereditarily Baire hyperspace. We do not know if such a space can be constructed in ZFC.},
author = {Banakh, Taras, Plichko, Anatolij},
journal = {RACSAM},
keywords = {linear metric spaces; Baire properties; algebraic dimension},
language = {eng},
number = {1-2},
pages = {31-37},
title = {The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.},
url = {http://eudml.org/doc/41640},
volume = {100},
year = {2006},
}

TY - JOUR
AU - Banakh, Taras
AU - Plichko, Anatolij
TI - The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.
JO - RACSAM
PY - 2006
VL - 100
IS - 1-2
SP - 31
EP - 37
AB - Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space X with meager hyperspace K(X). Also under (d = c) we construct a metrizable separable noncomplete linear metric space with hereditarily Baire hyperspace. We do not know if such a space can be constructed in ZFC.
LA - eng
KW - linear metric spaces; Baire properties; algebraic dimension
UR - http://eudml.org/doc/41640
ER -

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