Definition:Real Interval/Half-Open

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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 defined as

Some sources, when defining a half-open real interval, require that $a < b$.

This is to eliminate the degenerate case where the interval is the empty set.


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.

Some sources call it a partly open interval or partly 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


Sources


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