A second category set with only first category functions

Komjáth, P.

doi:10.1090/S0002-9939-1991-1065086-7

A second category set with only first category functions
HTML articles powered by AMS MathViewer

by P. Komjáth

Proc. Amer. Math. Soc. 112 (1991), 1129-1136

DOI: https://doi.org/10.1090/S0002-9939-1991-1065086-7

PDF | Request permission

Abstract:

If the existence of a measurable cardinal is consistent then it is consistent in that there is a second category set $A \subseteq R$ such that every $A \to A$ function, as a subset of ${R^2}$, is of first category. Some other connected results are also proved.

References

William Boos, Lectures on large cardinal axioms, $\vDash$ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974) Lecture Notes in Math., Vol. 499, Springer, Berlin, 1975, pp. 25–88. MR 0414364
Hans-Dieter Donder, Regularity of ultrafilters and the core model, Israel J. Math. 63 (1988), no. 3, 289–322. MR 969944, DOI 10.1007/BF02778036
P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79), no. 2, 113–138. MR 533932, DOI 10.2307/44151159
P. Erdős and P. Komjáth, Countable decompositions of $\textbf {R}^2$ and $\textbf {R}^3$, Discrete Comput. Geom. 5 (1990), no. 4, 325–331. MR 1043714, DOI 10.1007/BF02187793
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
P. Komjáth, On second-category sets, Proc. Amer. Math. Soc. 107 (1989), no. 3, 653–654. MR 976358, DOI 10.1090/S0002-9939-1989-0976358-7
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, DOI 10.2307/2271949
Menachem Magidor, On the existence of nonregular ultrafilters and the cardinality of ultrapowers, Trans. Amer. Math. Soc. 249 (1979), no. 1, 97–111. MR 526312, DOI 10.1090/S0002-9947-1979-0526312-2

Sur un théorème équivalent à l’hypothèse du continu

Sierpinśki’s theorem

Über gewisse Zerlegungen von Mengen

20