Proposition 2

The multiplicative group $\mathbf{R}^*$ of real numbers $\neq 0$ is a topological group isomorphic to the product of its subgroups $\mathbf{R}_+^*$ and $\mathbf{U}_0$, where

For each $x \neq 0$ let $\operatorname{sgn} x$ denote $\frac{x}{|x|}$ (sign of $x$). The function $\operatorname{sgn}$ is a homomorphism of $\mathbf{R}^*$ onto $\mathbf{U}_0$. We have $x = |x|\operatorname{sgn} x$, and this decomposition of $x$ as the product of an element of $\mathbf{R}_+^*$ and an element of $\mathbf{U}_0$ is unique; hence the group structure of $\mathbf{R}^*$ is the product of the group structures of $\mathbf{R}_+^*$ and $\mathbf{U}_0$. On the other hand, the mapping $x \to |x|$ is continuous, and so is $x \to \operatorname{sgn} x = \frac{x}{|x|}$, since $x \neq 0$. Hence the result. We extend the function $\operatorname{sgn}$ to the whole of $\mathbf{R}$ by putting $\operatorname{sgn} 0 = 0$.

We shall see in Chapter V (§ 4, no. 1, Theorem 1) that the topological group $\mathbf{R}_+^*$ is isomorphic to the additive group $\mathbf{R}$; this will complete the determination of the structure of the topological group $\mathbf{R}_*$.