intervals and segments

An interval or a segment in $\mathbb{R}$ means the set of every points $x$ between $a\in \mathbb{R}$ and $b\in \mathbb{R}$ , ($a<b$).

We shall define intervals and segments in $\mathbb{R}$ accuratelly.

If the set does not include both endpoints, it is called the open interval or the segment. We write the open interval or the segment $(a,b)$ .

The interval or the closed interval which includes both endpoints is expressed $[a,b]$ .Furthermore, it is easy for you to understand the left half open interval $(a,b]$ and the right half open interval $[a,b)$ .

As $\pm\infty$  are not real numbers, the following intervals are impossible.
\[ [-\infty,a),\quad [-\infty,a],\quad (a,\infty],\quad [a,\infty],\quad [-\infty,\infty] \]
The intervals $(-\infty,a), (a,\infty), (-\infty,\infty)$  are regarded as open and $(-\infty,a], [a,\infty)$ are regarded as half open.