Remarks on a Theorem Of

In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and linear spaces. In general terms his theorem states that if J is a family of open maps defined in one topological space Xi into another, X2, the total image JiS) of a second category Baire set S in Xi has, under certain conditions on J and S, a nonvacuous interior. The point of these remarks is to show that his argument yields a theorem for a larger class than the second category Baire sets. From this there follow slightly stronger and more specific versions of some of his results, including his principal theorem, as well as a proof that if S is a subset of a weak sort of topological group and S contains a second category Baire set, then the identity element lies in the interior of both S~1S and SS^1. There is also at the end an extension of Zorn's theorem on the structure of certain semigroups. In a topological space X let the closure and interior of a set E be denoted by E* and E° and the null set by A. For any set S let 7(S) = U [G| C7 open, Gi^S is first category] and TT(S) =X — IiS), and let IlliS) be the open set 77(S)on7(X-S). By a fundamental theorem of Banach [2], SP\7(S)* is first category and hence S is second category if and only if 7/(S)°?iA. From these we note that if A7 is a non-null open subset of IlliS), then N—S is in the first category set iX — S)CMiX — S), and NC\S cannot be first category since N is non-null open and disjoint with 7(S). This gives us the following lemma.