On the convergence and character spectra of compact spaces

István Juhász ¹   ; William A. R. Weiss ²

¹ Alfréd Rényi Institute of Mathematics H-1053 Budapest, Hungary
² Department of Mathematics University of Toronto Toronto, Ontario, Canada M5S 2E4

Fundamenta Mathematicae, Tome 207 (2010) no. 2, pp. 179-196

Résumé

An infinite set $A$ in a space $X$ converges to a point $p$ (denoted by $A \to p$) if for every neighbourhood $U$ of $p$ we have $|A \setminus U| |A|.$ We call $cS(p,X) = \{|A| : A \subset X$ and $ A \to p\}$ the convergence spectrum of $p$ in $X$ and $cS(X) = \bigcup \{ cS(x,X) : x \in X \}$ the convergence spectrum of $X$. The character spectrum of a point $p \in X$ is $\chi S(p,X) = \{ \chi(p,Y) : p$ is non-isolated in $Y \subset X\}$, and $\chi S(X) = \bigcup \{ \chi S(x,X) : x \in X\}$ is the character spectrum of $X$. If $\kappa \in \chi S(p,X)$ for a compactum $X$ then $\{\kappa,{\rm cf}(\kappa)\} \subset cS(p,X)$.A selection of our results ($X$ is always a compactum):(1) If $\chi(p,X) > \lambda = \lambda^{ \widehat{t}(X)}$ then $\lambda \in \chi S(p,X)$; in particular, if $X$ is countably tight then $\chi(p,X) > \lambda = \lambda^\omega$ implies that $\lambda \in \chi S(p,X)$. (2) If $\chi(X) > 2^\omega$ then $\omega_1 \in \chi S(X)$ or $\{2^\omega, (2^\omega)^+\} \subset \chi S(X)$. (3) If $\chi(X) > \omega$ then $\chi S(X) \cap [\omega_1,2^\omega] \ne \emptyset$. (4) If $\chi(X) > 2^\kappa$ then $\kappa^+ \in cS(X)$, in fact there is a converging discrete set of size $\kappa^+$ in $X$. (5) If we add $\lambda$ Cohen reals to a model of GCH then in the extension for every $\kappa \le \lambda$ there is $X$ with $\chi S(X) = \{\omega,\kappa\}$. In particular, it is consistent to have $X$ with $\chi S(X) = \{\omega, \aleph_\omega\}$. (6) If all members of $\chi S(X)$ are limit cardinals then $|X| \le (\sup \{|\overline{S}|: S \in [X]^\omega\})^\omega.$ (7) It is consistent that $2^\omega$ is as big as you wish and there are arbitrarily large $X$ with $\chi S(X) \cap (\omega,2^\omega) = \emptyset$.It remains an open question if, for all $X$, $\min cS(X) \le \omega_1$ (or even $\min \chi S(X) \le \omega_1$) is provable in ZFC.

Détail

DOI : 10.4064/fm207-2-6

Keywords: infinite set space converges point denoted every neighbourhood have setminus call subset convergence spectrum bigcup convergence spectrum character spectrum point chi chi non isolated subset chi bigcup chi character spectrum nbsp kappa chi compactum kappa kappa subset selection results always compactum chi lambda lambda widehat lambda chi particular countably tight chi lambda lambda omega implies lambda chi chi omega omega chi omega omega subset chi chi omega chi cap omega omega emptyset chi kappa kappa there converging discrete set size kappa lambda cohen reals model gch extension every kappa lambda there chi omega kappa particular consistent have chi omega aleph omega members chi limit cardinals sup overline omega omega consistent omega you wish there arbitrarily large chi cap omega omega emptyset remains question min omega even min chi omega provable zfc