Automatic continuity of group operations
A subset \(A\) of a topological space \(X\) is said to have the Baire property if there is an open set \(U\subseteq X\) with \(A\Delta U\) meagre. Call a function \(f: C\to Y\) Baire measurable if, for any open set \(U\subseteq Y\), \(f^{-1}(U)\) has the Baire property. It is proved that if \((G,\cdot)\) is a group with a metric, separable and Baire topology \(\tau\) such that \(h\to g\cdot g\) is continuous for all \(g\in G\) and \(g\to g\cdot h\) is Baire measurable for all \(h\in G\), then \((G,\tau)\) is a topological group. As a consequence of this result, the author also considers the standard Borel groups. Some results of \textit{A. S. Kechris} [Classical descriptive set theory. Graduate Texts in Math. 156 (Berlin 1995; Zbl 0819.04002)] concerning Polish groups and free actions of standard Borel groups are established.
scientific article; zbMATH DE number 1022201