Partitions of topological spaces and a new club-like principle

Carvalho, Rodrigo; Fernandes, Gabriel; Junqueira, Lúcia

doi:10.1090/proc/16208

Partitions of topological spaces and a new club-like principle
HTML articles powered by AMS MathViewer

by Rodrigo Carvalho, Gabriel Fernandes and Lúcia R. Junqueira

Proc. Amer. Math. Soc. 151 (2023), 1787-1800

DOI: https://doi.org/10.1090/proc/16208

Published electronically: January 13, 2023

HTML | PDF | Request permission

Abstract:

We give a new proof of the following theorem due to W. Weiss and P. Komjath: if $X$ is a regular topological space, with character $< \mathfrak {b}$ and $X \rightarrow (top\, \omega + 1)^{1}_{\omega }$, then, for all $\alpha < \omega _1$, $X \rightarrow (top\, \alpha )^{1}_{\omega }$, fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle $\clubsuit _{F}$, and use it to prove that it is consistent with $\neg CH$ that $\mathfrak {b}$ is the optimal bound for the character of $X$. In [Proc. Amer. Math. Soc. 101 (1987), pp. 767–770], this was obtained using $\diamondsuit$.

References

William Chen, Variations of the stick principle, Eur. J. Math. 3 (2017), no. 3, 650–658. MR 3687435, DOI 10.1007/s40879-017-0167-z
Keith J. Devlin, Constructibility, Perspectives in Logic, Cambridge University Press, 2017.
Harvey Friedman, On closed sets of ordinals, Proc. Amer. Math. Soc. 43 (1974), 190–192. MR 327521, DOI 10.1090/S0002-9939-1974-0327521-9
Harvey Friedman, One hundred and two problems in mathematical logic, The Journal of Symbolic Logic 40 (1975), no. 2, 113–129.
Sakaé Fuchino, Saharon Shelah, and Lajos Soukup, Sticks and clubs, Ann. Pure Appl. Logic 90 (1997), no. 1-3, 57–77. MR 1489304, DOI 10.1016/S0168-0072(97)00030-4
P. Komjáth and W. Weiss, Partitioning topological spaces into countably many pieces, Proc. Amer. Math. Soc. 101 (1987), no. 4, 767–770. MR 911048, DOI 10.1090/S0002-9939-1987-0911048-6
J. Nešetřil and V. Rödl, Ramsey topological spaces, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976) Soc. Czech. Math. and Physicists, Prague, 1977, pp. 333–337. MR 458361
J. Nešetřil and V. Rödl, Partition theory and its application, Surveys in Combinatorics. Cambridge University Press, Cambridge, 1979, pp. 96–156.
Karel Prikry and Robert M. Solovay, On partitions into stationary sets, J. Symbolic Logic 40 (1975), no. 1, 75–80. MR 432455, DOI 10.2307/2272274
William Weiss, Partitioning topological spaces, Mathematics of Ramsey theory, Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 154–171. MR 1083599, DOI 10.1007/978-3-642-72905-8_{1}1