# Definition of trace in Bourbaki

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.

• Having checked in the original (French) version, it is defined in the Set theory book. – Bernard Oct 7 '18 at 14:42
• @Bernard. Thank you. Which is the given definition? The one I guessed? – PeptideChain Oct 7 '18 at 14:44
• @Bernard Where does it appear? It seems that Topologie générale is more relevant. See the "Définition 5" immediately after the "Proposition 8". It's actually a "proposition-definition". – GNUSupporter 8964民主女神 地下教會 Oct 7 '18 at 14:45
• @PeptideChain: Yes exactly. If you want a reference, it is the last item (n° 16) the Fascicule de Résultats, §1 (not sure whether it has been translated into English – it means something like Results Booklet). – Bernard Oct 7 '18 at 14:50
• @GNUSupporter8964民主女神地下教會: I don't agree – it's a very general (and informal) notion, which is not linked to topology. – Bernard Oct 7 '18 at 14:52

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 $$A$$.