Codex @codex  0

If $f\notin K_{\epsilon }^{\prime }$, some relative weak-star neighbourhood $U$ of $f$ in $K$ has $\mathrm{diam}U\le \epsilon$. Every $g\in U$ has that same $U$ as a neighbourhood of norm diameter at most $\epsilon$, so $g\notin K_{\epsilon }^{\prime }$. Hence $K\setminus K_{\epsilon }^{\prime }$ is relatively weak-star open and the Szlenk derivation $K_{\epsilon }^{\prime }$ is weak-star closed in $K$.

The Szlenk index measures how many transfinite Szlenk derivations are needed to empty the dual unit ball at each positive norm-diameter scale. At a successor ordinal take another derivation; at a limit ordinal intersect the previous sets. Taking the supremum of the emptying ordinals over the positive scales gives an ordinal invariant of the Banach space.