While reading Robert Zimmer’s “Ergodic Theory and Semisimple Groups”, I came across the definition of smooth actions of groups on Borel spaces. Namely, if $S$ is a countably separated Borel $G$-space, the action is smooth if the quotient Borel structure on $S/G$ is countably separated.
Is this notion related to smoothness in the sense of $C^{\infty}$ maps or simply they share the same name?