Universally Kuratowski–Ulam spaces

David Fremlin; Tomasz Natkaniec; Ireneusz Recław

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 3, page 239-247
  • ISSN: 0016-2736

Abstract

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We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair (X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X× Y. A space Y is called a universally Kuratowski-Ulam (uK-U) space if (X,Y) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable π-basis is uK-U. We prove the following:  • every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uK-U (so there are uK-U Baire spaces which do not have countable π-bases);  • every Baire uK-U space is ccc.

How to cite

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Fremlin, David, Natkaniec, Tomasz, and Recław, Ireneusz. "Universally Kuratowski–Ulam spaces." Fundamenta Mathematicae 165.3 (2000): 239-247. <http://eudml.org/doc/212468>.

@article{Fremlin2000,
abstract = { We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair (X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X× Y. A space Y is called a universally Kuratowski-Ulam (uK-U) space if (X,Y) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable π-basis is uK-U. We prove the following:  • every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uK-U (so there are uK-U Baire spaces which do not have countable π-bases);  • every Baire uK-U space is ccc. },
author = {Fremlin, David, Natkaniec, Tomasz, Recław, Ireneusz},
journal = {Fundamenta Mathematicae},
keywords = {Baire space; dyadic space; quasi-dyadic space; Kuratowski-Ulam Theorem; Kuratowski-Ulam pair; universally Kuratowski-Ulam space; Kuratowski-Ulam theorem},
language = {eng},
number = {3},
pages = {239-247},
title = {Universally Kuratowski–Ulam spaces},
url = {http://eudml.org/doc/212468},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Fremlin, David
AU - Natkaniec, Tomasz
AU - Recław, Ireneusz
TI - Universally Kuratowski–Ulam spaces
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 3
SP - 239
EP - 247
AB - We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair (X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X× Y. A space Y is called a universally Kuratowski-Ulam (uK-U) space if (X,Y) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable π-basis is uK-U. We prove the following:  • every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uK-U (so there are uK-U Baire spaces which do not have countable π-bases);  • every Baire uK-U space is ccc.
LA - eng
KW - Baire space; dyadic space; quasi-dyadic space; Kuratowski-Ulam Theorem; Kuratowski-Ulam pair; universally Kuratowski-Ulam space; Kuratowski-Ulam theorem
UR - http://eudml.org/doc/212468
ER -

References

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  3. [FG] D. H. Fremlin and S. Grekas, Products of completion regular measures, Fund. Math. 147 (1995), 27-37. Zbl0843.28005
  4. [DF] D. H. Fremlin, Universally Kuratowski-Ulam spaces, preprint. 
  5. [HMC] R. C. Haworth and R. C. McCoy, Baire spaces, Dissertationes Math. 141 (1977). 
  6. [AK] A. S. Kechris, Classical Descriptive Set Theory, Springer, Berlin, 1995. 
  7. [KK] K. Kuratowski, Topologie I, PWN, Warszawa, 1958. 
  8. [ŁR] G. Łabędzki and M. Repický, Hechler reals, J. Symbolic Logic 60 (1995), 444-458. Zbl0832.03025
  9. [JO] J. Oxtoby, Measure and Category, Springer, 1980. 
  10. [IR] I. Recław, Fubini properties for σ-centered σ-ideals, Topology Appl., to appear. 
  11. [FT] F. D. Tall, The density topology, Pacific Math. J. 62 (1976), 275-284. 
  12. [HS] H. Steinhaus, Sur les distances des points des ensembles de mesure positive, Fund. Math. 1 (1920), 99-104. Zbl47.0179.02

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