Remark

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

General Topology, chapter III, § 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups

Let $H$ be any subgroup of a topological group $G$ . For each $x \in \overline{H}$ we have $x \overline{H} = x$ . $\overline{H} = \overline{H}$ , since translation by $x$ is a homeomorphism of $G$ onto $G$ . In other words, for each $x \in \overline{H}$ , $xH$ is dense in $\overline{H}$ . It follows that if $H$ is not closed, then $\overline{H} \cap CH$ is dense in $\overline{H}$ .

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

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Cites

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