Condensations of Cartesian products

Oleg I. Pavlov

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 2, page 355-365
  • ISSN: 0010-2628

Abstract

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We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space X there is a μ such that X μ can be condensed onto a normal ( σ -compact) space if and only if there is no measurable cardinal. For any Tychonoff space X and any cardinal ν there is a Tychonoff space M which preserves many properties of X and such that any one-to-one continuous image of M μ , μ ≤ ν , contains a closed copy of X μ . For any infinite compact space K there is a normal space X such that X × K cannot be mapped one-to-one onto a normal space.

How to cite

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Pavlov, Oleg I.. "Condensations of Cartesian products." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 355-365. <http://eudml.org/doc/248401>.

@article{Pavlov1999,
abstract = {We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space $X$ there is a $\mu $ such that $X^\mu $ can be condensed onto a normal ($\sigma $-compact) space if and only if there is no measurable cardinal. For any Tychonoff space $X$ and any cardinal $\nu $ there is a Tychonoff space $M$ which preserves many properties of $X$ and such that any one-to-one continuous image of $M^\mu $, $\mu \le \nu $, contains a closed copy of $X^\mu $. For any infinite compact space $K$ there is a normal space $X$ such that $X\times K$ cannot be mapped one-to-one onto a normal space.},
author = {Pavlov, Oleg I.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {condensation; one-to-one; compact; measurable; topologies on product; compact space; measurable cardinal},
language = {eng},
number = {2},
pages = {355-365},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Condensations of Cartesian products},
url = {http://eudml.org/doc/248401},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Pavlov, Oleg I.
TI - Condensations of Cartesian products
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 355
EP - 365
AB - We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space $X$ there is a $\mu $ such that $X^\mu $ can be condensed onto a normal ($\sigma $-compact) space if and only if there is no measurable cardinal. For any Tychonoff space $X$ and any cardinal $\nu $ there is a Tychonoff space $M$ which preserves many properties of $X$ and such that any one-to-one continuous image of $M^\mu $, $\mu \le \nu $, contains a closed copy of $X^\mu $. For any infinite compact space $K$ there is a normal space $X$ such that $X\times K$ cannot be mapped one-to-one onto a normal space.
LA - eng
KW - condensation; one-to-one; compact; measurable; topologies on product; compact space; measurable cardinal
UR - http://eudml.org/doc/248401
ER -

References

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  10. Yakivchik A.N., On tightenings of a product of finally compact spaces (in Russian), Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. 1989, no. 4, 84-86; translation in Moscow Univ. Math. Bull. 44.4 (1989), 86-88. MR1029765

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