0. THE ROLE OF COUNTABILITY IN (S) AND (L) In the first four sections of this monograph we shall show that "the unusual, and seemingly unnecessary, position of denumerability in topology" ([65], [64 p. 83]) appears to be quite essential in the statements of (S) and (L). We shall present there several examples of spaces showing that some high-level analogues of (S) and (L) are false. The examples will also show a strong influence of (S) and (L) on the set of real numbers. Let X be a space and let ~ be a reflexive and transitive relation on X. Then by X[~] we denote the space obtained by refining the topology on X by introducing the sets {yEX: y~x}, (xEX) as new open sets. The following is a list of few straightforward facts about X[~] which give us some indications which properties of X and ~ are needed in order to make X[~] relevant to (S) and (L). 0.0. LEMMA. The character of the space X[~] is not bigger than the character as X. PROOF. If B is an open neighborhood of x in X, then B(~ x] = {y E B : y ~ X} is an open neighborhood of x in X[~]. Moreover, if Bx is a basis of x in X, then {B[~ x]: BE Bx} is a basis of x in X[~]. 0.1. LEMMA. If X is regular (zero-dimensional) and if~ is closed in X2, then X[~] is also regular (zero-dimensional). PROOF. If~ is closed in X 2, then for all open B ~ X and x in X, B[~ x] is the intersection of an open set and a clopen set. Note also that the closure of B[~ x] in X[~] is a subset of B[~ x]. Now the conclusion is immediate. 3 http://dx.doi.org/10.1090/conm/084
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