Interval

Simply, an interval is a connected portion of the real line. Mathematically, a subset $I$ of the real line is called an interval if it contains at least two numbers and every number lying between them; that is, if $x,yϵI$ and $zϵ\mathbb{R},x<z<y$, then $zϵI$.

If we suppose that $a,bϵ\mathbb{R}$, we could have…

• an open interval: contains neither endpoint such that $(a,b)=\{xϵ\mathbb{R}:a<x<b\}$.
• a closed interval: contains both endpoints such that $[a,b]=\{xϵ\mathbb{R}:a\le x\le b\}$.
• a half-open interval: contains one endpoint such that $(a,b]=\{xϵ\mathbb{R}:a<x\le b\}$.

These could be finite or infinite, or semi-infinite: For a real value $aϵ\mathbb{R}$, we could have $(a,\infty )=\{xϵ\mathbb{R}:x>a\}$ or $[a,\infty )=\{xϵ\mathbb{R}:x\ge a\}$. This can be the same for $-\infty$. Note that there is no square bracket on $\infty$, as infinity is not a number and so can never be included. All of those examples are semi-infinite.