Definition of trace in Bourbaki

Question

Bourbaki, General Topology, p. 61 (1966)

What is the definition of trace in the following Proposition?

Proposition 8. Let $\mathcal{F}$ be a filter on a set $X$ and $A$ a subset of $X$. Then the trace $\mathcal{F}_A$ of $\mathcal{F}$ on $A$ is a filter if and only if each set of $\mathcal{F}$ meets $A$.

My guess:

Definition. Given a filter $\mathcal{F}$ on $X$ and a subset $x\subseteq X$, the trace of $\mathcal{F}$ on $x$ is denoted and defined by $$ \mathcal{F}_x:=\{x\cap y:y\in\mathcal{F}\}. $$

I have not found the definition in the book, in any case there is no trace of the trace in the index.

Possibly this definition is fallen into disuse, because some online search gave me no result.

Answer 1

The definition that you're after is taken from Bourbaki's Theory of Sets (chapter II, section 4, subsection 5):

The intersection $X\cap A$ is sometimes called the trace of $X$ on $A$. If $\mathcal F$ is a family of sets, the set of traces on $A$ of the sets belonging to $\mathcal F$ is called the trace of $\mathcal F$ on $A$.