A Metrization Theorem for Diagonal Calibres

In my previous talk, we saw that if $X$ is a compact space such that $X^2/D$ has relative calibre $(\omega_1,\omega)$ in $K(X^2/D)$, then $X$ need not be metrizable. Here, $D$ is the diagonal in $X^2$ and $K(Y)$ denotes the set of compact subspaces of Y. In this talk, we will show that the stronger property that $K(X^2/D)$ has calibre $(\omega_1,\omega)$ does, in fact, imply metrizability for compact spaces $X$.