Proposition 2

Every homeomorphism $f$ of $\mathbf{R}$ onto an interval ]$a, b[$ can be extended to a homeomorphism $\overline{f}$ of $\overline{\mathbf{R}}$ onto $[a, b]$. If $f$ is an increasing function, then $\overline{f}$ is an order isomorphism of $\overline{\mathbf{R}}$ onto $[a, b]$.

Let $f$ be an increasing homeomorphism. If we extend $f$ to $\mathbf{R}$ by putting $\overline{f}(-\infty) = a$ and $\overline{f}(+\infty) = b$, it is obvious that $\overline{f}$ is a strictly increasing mapping (and therefore a bijection) of $\overline{\mathbf{R}}$ onto $[a, b]$. Hence $\overline{f}$ maps every open interval of $\overline{\mathbf{R}}$ onto an interval which is open with respect to $[a, b]$, and is therefore a homeomorphism of $\mathbf{R}$ onto $[a, b]$ by reason of Definition 1 and Proposition 5 of § 1, no. 4.

If $f$ is decreasing, we apply what has been proved to the increasing homeomorphism $x \to -f(x)$ of $\mathbf{R}$ onto ]$-b, -a[$.

All the properties of the interval $[a, b]$ obtained in § 2, which involve only the order structure and the topology of the interval, can therefore be transported to $\overline{\mathbf{R}}$; hence the following propositions: