A family A⊆𝒫(ω) is called countably splitting if for every countable F⊆[ω]ω, some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an Fσ splitting family that does not contain a closed splitting family.
Otmar Spinas. "Analytic countably splitting families." J. Symbolic Logic 69 (1) 101 - 117, March 2004. https://doi.org/10.2178/jsl/1080938830