Baire irresolvable spaces and ideal theory
Abstract
In the New Scottish Book M. Katĕtov asked whether there exists a Hausdorff space X without isolated points such that every real-valued function on X is continuous at some point? In the paper it is shown that the existence of such a space is equiconsistent to the existence of measurable cardinal.
References
1. J. Baumgartner, A. Taylor, S. Wagon, Structural properties of ideals, (to appear in Dissertationes Math.).
2. A.G. Elkin, Ultrafilters and irresolvable spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 24 (1969), 51-56.
3. A.M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489.
4. E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333.
5. T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 165-198.
6. T. Jech, Set Theory, Academic Press 1978.
7. T. Jech, M. Magidor, W. Mitchell, K. Prikry, Precipitous ideals, (to appear).
8. T. Jech, K. Prikry, Ideals over uncountable sets: Applications of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc. 214.
9. M. Katĕtov, On topological spaces containing no disjoint dense sets, Mat. Sb. 21 (1947), 3-12.
10. K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227.
11. K. Kunen, Saturated ideals, J. Symbolic Logic, 43 (1978), 65-76.
12. K. Kunen, Maximal ?-Independent Families, Technical Report # 15, Univer. of Texas at Austin, Dec. 1979.
13. D. Maharam, On a theorem of von Neumann, Proc. Amer. Math. Soc. 9 (1958), 987-994.
14. V.I. Malyhin, On the resolvability of the product of two spaces and a problem of Katětov, Dokl. Akad. Nauk SSSR 222 (1975), 725-729.
15. R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56.
16. R. Solovay, Real-valued measurable cardinals, Axiomatic Set Theory, Proc. Sympos. Pure Math. 13 (1971), 397-428.
17. A. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), 33-55.
18. A. Taylor, Diamond principles, ideals and the normal Moore space problem, Canad. J. Math., (to appear).
19. A. Ionescu Tulcea, C. Ionescu Tulcea, Topics in the theory of liftings, Ergebnisse der Math. 48, Springer-Verlag, Berlin 1969.
20. S. Wagon, The structure of precipitous ideals, Fund. Math. 106 (1980), 47-52.
21. B. Węglorz, Some properties of filters, (preprint).
22. H. Woodin, An ℵ_1-dense ℵ_1-complete ideal on ℵ_1, (preprint).
2. A.G. Elkin, Ultrafilters and irresolvable spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 24 (1969), 51-56.
3. A.M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489.
4. E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333.
5. T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 165-198.
6. T. Jech, Set Theory, Academic Press 1978.
7. T. Jech, M. Magidor, W. Mitchell, K. Prikry, Precipitous ideals, (to appear).
8. T. Jech, K. Prikry, Ideals over uncountable sets: Applications of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc. 214.
9. M. Katĕtov, On topological spaces containing no disjoint dense sets, Mat. Sb. 21 (1947), 3-12.
10. K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227.
11. K. Kunen, Saturated ideals, J. Symbolic Logic, 43 (1978), 65-76.
12. K. Kunen, Maximal ?-Independent Families, Technical Report # 15, Univer. of Texas at Austin, Dec. 1979.
13. D. Maharam, On a theorem of von Neumann, Proc. Amer. Math. Soc. 9 (1958), 987-994.
14. V.I. Malyhin, On the resolvability of the product of two spaces and a problem of Katětov, Dokl. Akad. Nauk SSSR 222 (1975), 725-729.
15. R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56.
16. R. Solovay, Real-valued measurable cardinals, Axiomatic Set Theory, Proc. Sympos. Pure Math. 13 (1971), 397-428.
17. A. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), 33-55.
18. A. Taylor, Diamond principles, ideals and the normal Moore space problem, Canad. J. Math., (to appear).
19. A. Ionescu Tulcea, C. Ionescu Tulcea, Topics in the theory of liftings, Ergebnisse der Math. 48, Springer-Verlag, Berlin 1969.
20. S. Wagon, The structure of precipitous ideals, Fund. Math. 106 (1980), 47-52.
21. B. Węglorz, Some properties of filters, (preprint).
22. H. Woodin, An ℵ_1-dense ℵ_1-complete ideal on ℵ_1, (preprint).
KunenK., SzymańskiA., & TallF. (1986). Baire irresolvable spaces and ideal theory. Annales Mathematicae Silesianae, 2, 98-107. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/14330
Kenneth Kunen
Department of Mathematics, University Wisconsin, U.S.A. United States
Department of Mathematics, University Wisconsin, U.S.A. United States
Andrzej Szymański
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Franklin Tall
Department of Mathematics, University of Toronto, Canada Canada
Department of Mathematics, University of Toronto, Canada Canada
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