Large strongly antiUrysohn spaces exist
Abstract
As defined in [1], a Hausdorff space is strongly antiUrysohn (in short: SAU) if it has at least two nonisolated points and any two infinite} closed subsets of it intersect. Our main result answers the two main questions of [1] by providing a ZFC construction of a locally countable SAU space of cardinality $2^{\mathfrak{c}}$. The construction hinges on the existence of $2^{\mathfrak{c}}$ weak Ppoints in $\omega^*$, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality $> 2^{\mathfrak{c}}$ could exist, while it was shown in [1] that $2^{2^{\mathfrak{c}}}$ is an upper bound. Also, we do not know if crowded SAU spaces, i.e. ones without any isolated points, exist in ZFC but we obtained the following consistency results concerning such spaces. (1) It is consistent that $\mathfrak{c}$ is as large as you wish and there is a locally countable and crowded SAU space of cardinality $\mathfrak{c}^+$. (2) It is consistent that both $\mathfrak{c}$ and $2^\mathfrak{c}$ are as large as you wish and there is a crowded SAU space of cardinality $2^\mathfrak{c}$. (3) For any uncountable cardinal ${\kappa}$ the following statements are equivalent: (i) ${\kappa}=cof({[{\kappa}]}^{\omega},\subseteq)$. (ii) There is a locally countable and crowded SAU space of size ${\kappa}$ in the generic extension obtained by adding $\kappa$ Cohen reals. (iii) There is a locally countable and countably compact $T_1$space of size ${\kappa}$ in some CCC generic extension. [1] I. Juhasz, L. Soukup, and Z. Szentmiklossy, AntiUrysohn spaces, Top. Appl., 213 (2016), pp. 823.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.00618
 arXiv:
 arXiv:2106.00618
 Bibcode:
 2021arXiv210600618J
 Keywords:

 Mathematics  General Topology;
 54A25;
 54A35;
 54D10;
 03E04
 EPrint:
 18 pages