A Counterexample for Metrizability

A partially ordered set $P$ is said to have calibre $(\omega_1,\omega)$ if every uncountable subset of $P$ contains an infinite subset with an upper bound in $P$. It is conjectured that if $X$ is a compact space such that $K(X^2/D)$ has calibre $(\omega_1,\omega)$, where $D$ is the diagonal in $X^2$ and $K(Y)$ denotes the hyperspace of compact subsets of $Y$, then $X$ is metrizable. In this talk, we show that the weaker condition that a compactum $X$ has relative calibre $(\omega_1,\omega)$ in $K(X^2/D)$ does not imply metrizability of $X$. The example exploits a technique of van Douwen.