On the countable definability of sets

LetQ be a connected set in ^p. Denote byD[Q] the set of all domains containingQ, and letW(Q) be the set of all convex domains fromD[Q]. We present tests for classesD[Q] andW(Q) (in the case whenQ is convex for the last one) to have a countable basis. The results are expressed in terms of properties of the boundary FrQ of the setQ.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 382–395, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-242.