Analytic infinite gaps

We provide infinite-dimensional versions of analytic gap dichotomies, in the sense that a sequence of analytic hereditary families {Ip }p<ω of subsets of a countable set Ω is either countably separated or there is a tree structure inside Ω in which p-chains are sets from Ip . A topological version of this is that if K is a separable Rosenthal compact space, then either K is a continuous image of a finite-to-one preimage of a metric compactum or there is a tree structure inside K in which p-chains inside every branch form a relatively discrete family of sets.