On the convergence and character spectra of compact spaces
Tom 207 / 2010
Streszczenie
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.
