Proposition 2

Let G be a connected compact group.

a$) C(G)_0$ is a commutative connected compact group; D(G) is a connected compact group, equal to its derived group.

b) The continuous homomorphism $(x, y) \rightarrow xy$ from $C(G)_0\times D(G)$ to G is surjective and its kernel is a central subgroup of $C(G)_0\times D(G)$ that is compact and totally discontinuous.

c) There exists a family $(S_{\lambda})_{\lambda\in L}$ of almost simple compact Lie groups and a surjective continuous homomorphism $\prod_{\lambda\in L}S_{\lambda}\rightarrow D(G)$, whose kernel is a

totally discontinuous, compact, central subgroup.

Let $(G_{\alpha}, f_{\alpha \beta})$ be a projective system of compact Lie groups, relative to a filtered set I, such that G is isomorphic to lim$_{\leftarrow -}G_{\alpha}$ and such that the canonical maps $f_{\alpha}: G\rightarrow G_{\alpha}$ are surjective (Cor. 2 of Prop. 1). For $\alpha \in I$, let $\pi_{\alpha}: \widetilde{D}(G_{\alpha})\rightarrow D(G_{\alpha})$ be a universal covering of the group $D(G_{\alpha})$. The $f_{\alpha \beta}$ induce morphisms $\widetilde{f}_{\alpha \beta}: \widetilde{D}(G_{\beta})\rightarrow \widetilde{D}(G_{\alpha})$, and $( \widetilde{D}(G_{\alpha}),\widetilde{f}_{\alpha \beta})$ is a projective system of topological groups satisfying the hypotheses of Lemma 3.

It follows from this lemma that the topological group lim$_{\leftarrow -}\widetilde{D}(G_{\alpha})$ is isomorphic to the product of a family $(S_{\lambda})_{\lambda\in L}$ of almost simple compact Lie groups. By Lemma 1, the limit of the projective system of homomorphisms $(\pi_{\alpha})$ can be identified with a continuous homomorphism $\pi :\prod_{\lambda\in L}S_{\lambda}\rightarrow D(G)$,

which is surjective (General Topology, Chap. I, §9, no. 6, Cor. 2 of Prop. 8).

Now observe that the group $\prod_{\lambda\in L}S_{\lambda}$ is equal to its derived group: this

follows from §4, no. 5, Cor. of Prop. 10. The same is true for D(G), since $\pi$ is surjective. Consequently, $D(G)\supset D(D(G)) = D(G)$. Thus, the group D(G) is compact and equal to its derived group; this proves a), since the assertions concerning $C(G)_0$ are trivial.

On the other hand, the kernel of $\pi :\prod_{\lambda\in L}S_{\lambda}\rightarrow D(G)$ can be identified

with lim$_{\leftarrow -}$ Ker($\pi_{\alpha}$) $($Algebra, Chap. II, §6, no. 1, Remark 1), and thus with a

compact, totally discontinuous, central subgroup, hence c).

We prove b). For all $\alpha$ in I, the morphism $s_{\alpha}: C(G_{\alpha})_0\times D(G_{\alpha})\rightarrow G_{\alpha}$ such that $s_{\alpha}(x, y) =xy$ for $x\in C(G_{\alpha})_0, y\in D(G_{\alpha})$, is surjective and its kernel is a finite central subgroup (§1, no. 4, Cor. 1 of Prop. 4). The $s_{\alpha}$ form a projective system of maps, whose limit can, by the preceding, be identified with the homomorphism $(x, y) \rightarrow xy$ from $C(G)_0\times D(G)$ to G. We now see as before that this map is surjective and that its kernel is central and totally discontinuous, hence b).