A (real) interval is a set of real numbers between two other numbers. Intervals of ${\displaystyle x}$ are commonly used in graphs. An open interval excludes the end points, while the end points of a closed interval are elements. Some intervals may partially open and closed. 

An (integer) interval is basically the same as a real interval, except it consists of integers.

Formal definitions

Real interval

• Open interval: ${\displaystyle (a,b)=\{x\in\R:a<x<b\}}$
• Closed interval: ${\displaystyle [a,b]=\{x\in\R:a\le x\le b\}}$
• Left open, right closed: ${\displaystyle (a,b]=\{x\in\R:a<x\le b\}}$
• Right closed, right open: ${\displaystyle [a,b)=\{x\in\R:a\le x<b\}}$

Integer interval

• Open interval: ${\displaystyle (a,b)=\{x\in\Z:a<x<b\}}$
• Closed interval: ${\displaystyle [a,b]=\{x\in\Z:a\le x\le b\}}$
• Left open, right closed: ${\displaystyle (a,b]=\{x\in\Z:a<x\le b\}}$
• Right closed, right open: ${\displaystyle [a,b)=\{x\in\Z:a\le x<b\}}$

Arbitrary partially ordered set

Let ${\displaystyle (S,\le)}$ be a poset.

Let ${\displaystyle < }$ be the irreflexive kernel of ${\displaystyle \le}$

Then, an interval may be defined as:

• Open interval: ${\displaystyle (a,b)=\{x\in S:a<x<b\}}$
• Closed interval: ${\displaystyle [a,b]=\{x\in S:a\le x\le b\}}$
• Left open, right closed: ${\displaystyle (a,b]=\{x\in S:a<x\le b\}}$
• Right closed, right open: ${\displaystyle [a,b)=\{x\in S:a\le x<b\}}$