A topological version of $\diamondsuit$

Abstract:

The set-theoretic principle $\diamondsuit$ is shown to be equivalent to the existence of universal ${\omega _1}$-sequences in certain topological spaces. An ${\omega _1}$-sequence $({x_\alpha }:\alpha \in {\omega _1})$ in a space X is said to be universal in X if for every point $p \in X$ there is a stationary set $S \subseteq {\omega _1}$ so that the net $({x_\alpha }:\alpha \in S)$ converges to p. It is shown that the existence of universal ${\omega _1}$-sequences in spaces of weight $\leqslant c$ is equivalent to $\diamondsuit$.