Convergence With Respect to the Ideal of Meager Sets in Separable Category Bases

Convergence with respect to $F_{σ}$-supported ideals

“…It is well known that the space of Lebesgue measurable functions over R is a Fréchet space, whereas the space of all functions with the Baire property is not (cf. [7] and [4]).…”

“…Theorem 4 (cf. [4]). Convergence with respect to the σ-ideal K(X) of all meager sets in a second countable topological space X yields the Fréchet topology in the space of all real functions on X with the Baire property if and only if X = A ∪ B where A is an open set of the first category and B is the countable set of all isolated points of X.…”