Definition 5

Tag 02J2, permanent. Cite it as [Bourbaki, Tag 02J2].

Théories spectrales, chapter II, § 1. Transformation de Fourier

Let G, H and K be topological groups. Let $f: H\rightarrow G$ and $g: G\rightarrow K$ be morphisms of topological groups. The pair $(f, g)$ is said to be an exact sequence of topological groups if it is an exact sequence of groups (A, II, p. 10, Remark 5) and if $f$ and $g$ are strict morphisms.

An exact sequence will be represented by the diagram

H\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}K

and one will say that a diagram

G_1\longrightarrow^{f_1}G_2\longrightarrow^{f_2}G_3\rightarrow \cdots \rightarrow G_n\longrightarrow^{f_n}G_{n+1}

is exact if each pair $(f_i, f_{i+1})$ for $1\leqslant i\leqslant n-1$ is exact.

A sequence

1\rightarrow H\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}K\rightarrow 1

is exact if and only if $f$ is a strict injective morphism, $g$ is a strict surjective morphism, and the kernel of $g$ is equal to the image of $f$ . If K is separated, the image of $f$ is a closed subgroup of G.

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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