The Isbell function of $X$ is the map
\[ \mathsf{I}\colon \mathcal{P}\webleft (X\webright )\to \mathsf{Sets}\webleft (X,\mathcal{P}\webleft (X\webright )\webright ) \]
defined by
\[ \mathsf{I}\webleft (U\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[x\mapsto \webleft [U,\webleft\{ x\webright\} \webright ]_{X}]\mspace {-3mu}] \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
