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The notion of an A-nowhere dense set for various subsets A of the reals may prove to be of interest in its own right, but this paper will be concerned exclusively with the special case A = Q. Notice that rational perfect sets introduced by Miller in [1] form a subset of the Q-nowhere dense sets since the closure of a set is rational perfect if it is perfect and disjoint from the rationals. On the other hand, a set is Q-nowhere dense if its 2-sided-closure is disjoint from the rationals where the 2-sided-closure of a set refers to all those reals which are limits of both decreasing and increasing sequences from the set.

COVERING NUMBERS ASSOCIATED WITH TREES BRANCHING INTO A COUNTABLY GENERATED SET OF POSSIBILITIES