Bourbaki

Exercise 20, § 4

General Topology, III, 313

It is set in small type as supplementary.

Let φ\varphi be the canonical homomorphism R→T\mathbf{R} \to \mathbf{T} and let θ\theta be an irrational number. Define a group structure on the topological space R2×T2\mathbf{R}^2 \times \mathbf{T}^2 by the law

(x1,x2,t1,t2) (x1′,x2′,t1′,t2′)=(x1+x1′,x2+x2′,t1+t1′+φ(x2x1′),t2+t2′+φ(θx2x1′)). (x_1, x_2, t_1, t_2)\ (x'_1, x'_2, t'_1, t'_2) = (x_1 + x'_1, x_2 + x'_2, t_1 + t'_1 + \varphi(x_2 x'_1), t_2 + t'_2 + \varphi(\theta x_2 x'_1)).

G thus becomes a locally compact topological group (and even a Lie group). Show that the commutator subgroup of G is not closed. \*

Solution

No solution has been written yet.

Transcribed from the printed volume. Not translated.