Example 3

Tag 03V7, permanent. Cite it as [Bourbaki, Tag 03V7].

Theory of Sets, chapter IV, § 1. Structures and isomorphisms

As before, let $\mathscr{T}$ be the theory of sets, and consider the species of structures which has no auxiliary base set, one principal base set $\mathrm{A}$ , the typical characterization $\mathrm{V} \in \mathfrak{P}(\mathfrak{P}(\mathrm{A}))$ , and as axiom the transportable relation

(\forall \mathrm{V}) \, ((\mathrm{V}' \subset \mathrm{V}) \Rightarrow ((\bigcup_{\mathrm{X} \in \mathrm{V}'} \mathrm{X}) \in \mathrm{V}))

\text{and} \quad (\forall \mathrm{X})(\forall \mathrm{Y})((\mathrm{X} \in \mathrm{V} \text{ and } \mathrm{Y} \in \mathrm{V}) \Rightarrow ((\mathrm{X} \cap \mathrm{Y}) \in \mathrm{V})).

This species of structures is called the species of topological structures. A structure of this species is also called a topology, and the relation $\mathrm{X} \in \mathrm{V}$ is expressed by saying that $\mathrm{X}$ is an open set in the topology $\mathrm{V}$ (General Topology, Chapter I, § 1).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

*General Topology*, Chapter I, § 1