# Definition:Real Interval/Half-Open

< Definition:Real Interval(Redirected from Definition:Half-Open Real Interval)

## Contents

## Definition

Let $a, b \in \R$.

There are two **half-open (real) intervals** from $a$ to $b$.

#### Right half-open

The **right half-open (real) interval** from $a$ to $b$ is the subset:

- $\hointr a b := \set {x \in \R: a \le x < b}$

#### Left half-open

The **left half-open (real) interval** from $a$ to $b$ is the subset:

- $\hointl a b := \set {x \in \R: a < x \le b}$

## Also known as

This can often be seen rendered as **half open interval**.

## Examples

### Example $1$

Let $I$ be the unbounded closed real interval defined as:

- $I := \hointr 1 2$

Then $1 \in I$.

## Also see

- Definition:Open Real Interval
- Definition:Closed Real Interval
- Definition:Unbounded Open Real Interval
- Definition:Unbounded Closed Real Interval

- Definition:Half-Open Rectangle, a generalization to higher dimensional spaces

## Sources

- 1983: George F. Simmons:
*Introduction to Topology and Modern Analysis*... (previous) ... (next): $\S 1$: Sets and Set Inclusion