Lemma 1

Tag 0BXG, permanent. Cite it as [Bourbaki, Tag 0BXG].

General Topology, chapter VI, § 3. Real projective spaces

If every open (resp. closed) set in $A$ which is saturated with respect to $R_A$ is the trace on $A$ of an open (resp. closed) set in $E$ which is saturated with respect to $R$ , then the quotient space $A/R_A$ is homeomorphic to the subspace $f(A)$ of $E/R$ . This is so in particular if $A$ is open or closed in $E$ and saturated with respect to $R$ .

This follows from Proposition 10 of Chapter I, § 3, no. 6.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 3. Real projective spaces
Proposition 3, § 3
Proposition 5, § 3
Proposition 8, § 3

Cites

Proposition 10 of Chapter I, § 3, no. 6