Definition:Real Interval/Half-Open

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.

A half-open interval can also be referred to as a half-closed 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

• 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.1$ Definitions
• 1991: Felix Hausdorff: Set Theory (4th ed.) ... (previous) ... (next): Preliminary Remarks
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: open interval
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: open interval

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