Open subspaces and residual subspaces of Baire spaces are Baire

Given: The objects, hypotheses, and choice principles stated above.

For a topological space $X$, the following are equivalent: every countable intersection of dense open sets is dense; every countable union of closed sets with empty interior has empty interior; no nonempty open subset is meagre in $X$; and every residual subset meets every nonempty open set. The equivalence includes the empty space. (Equivalent forms of the Baire property).

For every topological space $X$, the meagre subsets of $X$ contain $\mathrm{\varnothing }$, are closed under taking subsets, and are closed under countable unions. (The meagre subsets of a topological space form a sigma-ideal).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space) and let $S\subseteq X$. The subspace topology (also relative topology) on $S$ is ${\mathcal{T}}_S:=\{U\cap S:U\in \mathcal{T}\},$ the family of traces on $S$ of the open sets of $X$. The pair $(S,{\mathcal{T}}_S)$ is a subspace of $X$. A subset of $S$ that lies in ${\mathcal{T}}_S$ is said to be open in $S$, and relatively open where the ambient space needs emphasis. (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).