Definition:Interval/Half-Open Interval

Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $a, b \in S$.

The left half-open interval between $a$ and $b$ is the set:

$\hointl a b := a^\succ \cap b^\preccurlyeq = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }$

where:

$a^\succ$ denotes the strict upper closure of $a$

$b^\preccurlyeq$ denotes the lower closure of $b$.

The right half-open interval between $a$ and $b$ is the set:

$\hointr a b := a^\succcurlyeq \cap b^\prec = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }$

where:

$a^\succcurlyeq$ denotes the upper closure of $a$

$b^\prec$ denotes the strict lower closure of $b$.

In the context of the real number line $\R$:

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

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

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