Covering Numbers Associated with Trees Branching into a Countably Generated Set of Possibilities

by Shelah. [Sh:660]
Real Analysis Exchange, 1998/99
This paper is concerned with certain generalizations of meagreness and their combinatorial equivalents. The simplest example, and the one which motivated further study in this area, comes about by considering the following definition: a set X subseteq R is said to be Q-nowhere dense if and only if for every rational q there exists and integer k such that the interval whose endpoints are q and q+1/k is disjoint from X . A set which is the union of countably many Q-nowhere dense sets will be called Q-very meagre. Steprans considered the least number of Q-meagre sets required to cover the real line and denoted by d_1 . He showed that there is a continuous function H --- first constructed by Lebesgue --- such that the least number of smooth functions into which H can be decomposed is equal to d_1 . This paper will further study d_1 and some of its generalizations. As well, an equivalence will be established between Q-meagreness and certain combinatorial properties of trees. This will lead to new cardinal invariants and various independence results about these will then be established.

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