Definition 1

The set $\overline{\mathbf{R}}$ endowed with the order structure and the topology defined above is called the extended real line.

When using the extended line $\overline{\mathbf{R}}$ it is often convenient to refer to its points as real numbers, by abuse of language; the points of $\mathbf{R}$ are then called finite real numbers. We shall adopt this convention in this section and the three following sections of this chapter; whenever we adopt this convention in future we shall indicate explicitly to which part of the text it extends.

If $a$ is a finite real number, the intervals $[a, +\infty[$ and ]—$\infty, a]$ (resp. ]$a, +\infty[$ and ]—$\infty, a[$) of $\overline{\mathbf{R}}$ are contained in $\mathbf{R}$ and coincide with the intervals of $\mathbf{R}$ hitherto denoted by $[a, \rightarrow[$ and ]$\leftarrow, a]$ (resp. ]$a, \rightarrow[$ and ]$\leftarrow, a[$); this new notation is much more often used. Again, $\mathbf{R}$ coincides with the interval ]—$\infty, +\infty[$ of $\overline{\mathbf{R}}$, and is sometimes so denoted.