Fragmentability of open sets and topological games (Q1985631)

Let \((X, \tau)\) be a regular topological space. The authors say that \((X, \tau)\) is a fos-space (space with fragmentable open sets) if there is a metric \(d \colon X \times X \to [0, \infty)\) such that, for every \(\varepsilon \in (0, \infty)\) and every nonempty \(A \in \tau\), there is \(U \in \tau\) such that \[ A \cap U \neq \varnothing \quad \text{and} \quad \sup\{d(x, y) \colon x, y \in A \cap U\} \leqslant \varepsilon. \] This notion can be considered as a weaker variant of the concept of fragmentable topological spaces introduced in [\textit{J. E. Jayne} and \textit{C. A. Rogers}, Acta Math. 155, 41--79 (1985; Zbl 0588.54020)]. The following theorem is proved. Theorem. The space \((X, \tau)\) is a fos-space if and only if it is the union of two disjoint sets \(X_1\) and \(X_2\) such that \(X_1\) is of the first Baire category in \(X\) and on \(X_2\) there exists a metric which generates a topology coarser than the one inherited from \((X, \tau)\). Moreover, a characterization of fos-spaces is given in the spirit of the Banach-Mazur game.