Proposition 2

Tag 0C2H, permanent. Cite it as [Bourbaki, Tag 0C2H].

General Topology, chapter IX, § 2. Metric spaces and metrizable spaces

The statements $d(x, A) = 0$ and $x \in \overline{A}$ are equivalent.

For $d(x, A) = 0$ expresses the fact that the ball $V_a(x)$ meets $A$ whatever the value of $a > 0$ ; and this is equivalent to $x \in \overline{A}$ .

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

Cited by

Remark, § 2
Proposition 7, § 2

Cites

Nothing.