Remarks on a theorem of E. J. McShane

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 7 is a family of open maps defined in one topological space X1 into another, X2, the total image 7(S) of a second category Baire set S in X1 has, under certain conditions on 7 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 sligh4ly 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 EO and the null set by A. For any set S let I(S) - U[GIG open, GnS is first category] and II(S)=X-I(S), and let III(S) be the open set II(S)0?I(X-S). By a fundamental theorem of Banach [2], SnI(S)* is first category and hence S is second category if and only if II(S)O7A. From these we note that if N is a non-null open subset of III(S), then N-S is in the first category set (X - S)G'iI(X - S), and NnQS cannot be first category since N is non-null open and disjoint with I(S). This gives us the following lemma. LEMMA 1. For any non-null open subset N of III(S), the sets N- S