Intervals.

$\displaystyle (a,b)$	$\displaystyle =\{ x \in \mathbb{R} \; \vert \; a<x<b \} \qquad ($ open interval) $\displaystyle .$
$\displaystyle [a,b]$	$\displaystyle =\{ x \in \mathbb{R} \; \vert \; a \leq x \leq b \} \qquad ($ closed interval $\displaystyle ).$
$\displaystyle (a,b]$	$\displaystyle =\{ x \in \mathbb{R} \; \vert \; a < x \leq b \}.$
$\displaystyle [a,b)$	$\displaystyle =\{ x \in \mathbb{R} \; \vert \; a \leq x<b \}.$
$\displaystyle (-\infty , b]$	$\displaystyle = \{ x \in \mathbb{R} \; \vert \; x \leq b \}.$
$\displaystyle (-\infty , b)$	$\displaystyle = \{ x \in \mathbb{R} \; \vert \; x < b \}.$
$\displaystyle [a, + \infty)$	$\displaystyle = \{ x \in \mathbb{R} \; \vert \; a \leq x \}.$
$\displaystyle (a, + \infty)$	$\displaystyle = \{ x \in \mathbb{R} \; \vert \; a<x \}.$
$\displaystyle (-\infty, + \infty )$	$\displaystyle = \mathbb{R}.$

Example 1.3.1

$\displaystyle \mathbb{R}- \{-1,3 \}$	$\displaystyle =(- \infty, -1 ) \cup (-1, 3) \cup (3, + \infty )$
$\displaystyle (2,6)$	$\displaystyle = \{ x \in \mathbb{R} \; ; \; 2 < x < 6 \}$ cf Fig. 1(a)
$\displaystyle \mathbb{R}^*$	$\displaystyle =(- \infty , 0) \cup (0,+ \infty )$
$\displaystyle [-1,+ \infty )$	$\displaystyle = \{ x \in \mathbb{R} \; ; \; x \geq -1 \}$ cf Fig. 1(b)
$\displaystyle \mathbb{R}^{+}$	$\displaystyle = [0, + \infty )$
$\displaystyle \mathbb{R}^{-}$	$\displaystyle = (- \infty , 0]$
$\displaystyle \mathbb{R}^{+*}$	$\displaystyle = (0, + \infty )$
$\displaystyle \mathbb{R}^{-*}$	$\displaystyle = (- \infty , 0)$

**Figure 1:** Intervals.
$\begin{figure}\mbox{ \subfigure[$(2,6) $]{\epsfig{file=OpenInterval.eps,width=6... ...[$[-1,+ \infty )$]{\epsfig{file=InfiniteInterv.eps,width=6cm} } }\end{figure}$