The property of Baire

A subset $A$ of a topological space $X$ has the property of Baire if there is an open $U\subseteq X$ such that

$A\mathrm{△}U=(A\setminus U)\cup (U\setminus A)$

is meagre in X, in the sense of Nowhere dense, meagre, residual, and comeagre subsets of a topological space. Meagre and comeagre refer to the ambient space X unless a relative space is explicitly named. No choice axiom or Baire-space hypothesis is part of this definition.

Every meagre set qualifies using $U=\mathrm{\varnothing }$, and every open set qualifies using itself, since the error is empty. In particular the empty set and the whole space qualify even if X is empty. If $M\subseteq {\bigcup }_n{N}_n$ with each $N_n$ nowhere dense, its complement contains ${\bigcap }_n(X\setminus \overline{N_n})$, a countable intersection of dense open sets. It need not contain a single dense open set. This definition uses the actual sequence of nowhere dense witnesses.