The rank of the diagonal and submetrizability

Aleksander V. Arhangel'skii; Raushan Z. Buzyakova

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 4, page 585-597
  • ISSN: 0010-2628

Abstract

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Several topological properties lying between the submetrizability and the G δ -diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular G δ -diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular G δ -diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable space. The rank 5 -diagonal plays a crucial role in this criterion. Every closed bounded subset of a Tychonoff space with a G δ -diagonal is shown to be Čech-complete. Under a slightly stronger condition, any such subset is shown to be a Moore space. We also establish that every closed bounded subset of a Tychonoff space with a regular G δ -diagonal is metrizable by a complete metric and, therefore, has the Baire property. Some further results are obtained, and new open problems are posed.

How to cite

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Arhangel'skii, Aleksander V., and Buzyakova, Raushan Z.. "The rank of the diagonal and submetrizability." Commentationes Mathematicae Universitatis Carolinae 47.4 (2006): 585-597. <http://eudml.org/doc/249865>.

@article{Arhangelskii2006,
abstract = {Several topological properties lying between the submetrizability and the $G_\delta $-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_\delta $-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_\delta $-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable space. The rank $5$-diagonal plays a crucial role in this criterion. Every closed bounded subset of a Tychonoff space with a $G_\delta $-diagonal is shown to be Čech-complete. Under a slightly stronger condition, any such subset is shown to be a Moore space. We also establish that every closed bounded subset of a Tychonoff space with a regular $G_\delta $-diagonal is metrizable by a complete metric and, therefore, has the Baire property. Some further results are obtained, and new open problems are posed.},
author = {Arhangel'skii, Aleksander V., Buzyakova, Raushan Z.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$G_\delta $-diagonal; rank $k$-diagonal; submetrizability; condensation; regular $G_\delta $-diagonal; zero-set diagonal; Čech-completeness; pseudocompact space; Moore space; Mrowka space; bounded subset; extent; Souslin number; -diagonal; rank -diagonal; submetrizability; condensation; regular -diagonal},
language = {eng},
number = {4},
pages = {585-597},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The rank of the diagonal and submetrizability},
url = {http://eudml.org/doc/249865},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Buzyakova, Raushan Z.
TI - The rank of the diagonal and submetrizability
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 4
SP - 585
EP - 597
AB - Several topological properties lying between the submetrizability and the $G_\delta $-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_\delta $-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_\delta $-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable space. The rank $5$-diagonal plays a crucial role in this criterion. Every closed bounded subset of a Tychonoff space with a $G_\delta $-diagonal is shown to be Čech-complete. Under a slightly stronger condition, any such subset is shown to be a Moore space. We also establish that every closed bounded subset of a Tychonoff space with a regular $G_\delta $-diagonal is metrizable by a complete metric and, therefore, has the Baire property. Some further results are obtained, and new open problems are posed.
LA - eng
KW - $G_\delta $-diagonal; rank $k$-diagonal; submetrizability; condensation; regular $G_\delta $-diagonal; zero-set diagonal; Čech-completeness; pseudocompact space; Moore space; Mrowka space; bounded subset; extent; Souslin number; -diagonal; rank -diagonal; submetrizability; condensation; regular -diagonal
UR - http://eudml.org/doc/249865
ER -

References

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