Question on Baire Property In reading Banach's book, Theory of Linear Operations, I have a question on the definition of Baire property, or Baire condition in Banach's book. Here is the definition in Banach's book:
Definition. A set $G$ is said to satisfy the Baire condition if every non-empty perfect set $P$ contains a point $x_0$ such that at least one of the sets $P \cap G$ and $P$ \ $G$ is of first category I at the point $x_0$ relative to $P$.
Note to say a set $G$ is of first category in a point $x_0$, it means that there exists a neighborhood $V$ of $x_0$, such that $G \cap V$ is of first category.
And the modern definition of Baire property from wikipedia:
Definition. A subset $A$ of the topological space $X$ is said to has the property of Baire if there exists an open set $U \subseteq X$ s.t. $A \Delta U$ is of first category.
I'm wondering whether these two definitions are equivalent. And why do all the closed set in metric space satisfy the Baire condition in Banach's definition.
 A: Kuratowski, in Topologie (part I) defines a set with the Baire property just as the Wikipedia does, although he states it as: a set $A$ has the property of Baire if there are open sets $U$ and sets of the first category $R$, $S$ such that $A = U + R - S$, which we would write now as $A = U \cup R \setminus S$. This corresponds to what you call the modern definition. 
He then shows that the following (pages 56,57 on the French edition that I have) are all to a set $A$, subset of $X$, having the property of Baire (with modern notation)
(here $D(B)$ for a subset $B$ is defined as the set of all points of $X$ at which $B$ is not of first category (i.e. of second category)):


*

*There exists a set of the first category $P$ such that $A \setminus P$ is a relatively closed-and-open set in $X \setminus P$.

*$A$ is the union of a $G_\delta$ and a first category subset of $X$.

*$A$ is the (set) difference between an $F_\sigma$ and a first category subset of $X$.

*$D(A) \cap D(X \setminus A)$ is nowhere dense. This means that every non-empty open set of $X$ has a point such that either $A$ or $X \setminus A$ is of first category at that point.

*$D(A) \setminus A$ is first category.
Point 4. in its alternative formulation seems pretty close to the definition that Banach gives. But there is also another notion, of being Baire in the restricted sense, which means that $A \cap P$ has the Baire property in every perfect set $P$ (in the subspace topology). This seems like Banach's definition. Kuratowski also gives it at page 60 ("proprieté de Baire au sens restreint"), and he shows it is a $\sigma$-algebra that contains all Borel sets. The notion is not equivalent to being Baire in the above sense, e.g. in this paper has a counterexample in $\mathbb{R}$, namely a Baire set that is not Baire in the restricted sense.
So in short: Banach defines $G$ to be "Baire in the restricted sense" which is equivalent to $G \cap E$ is Baire (standard sense) in $E$ (in the subspace topology), and which is stronger than being just a Baire set in the standard sense. But both are properties that all Borel sets have.   
