Definition:Interval/Ordered Set/Left Half-Open

Definition

Let $\struct {S, \preccurlyeq}$ be an 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 $b$.

Also defined as

Some sources, when defining a half-open interval, require that $a \preccurlyeq b$.

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

Also known as

A left half-open interval is also called:

a half-open on the left

a right half-closed interval

a half-closed interval on the right.

Also see

• Definition:Closed Interval
• Definition:Open Interval
• Definition:Right Half-Open Interval

• Definition:Left Half-Open Real Interval
• Definition:Right Half-Open Real Interval

• Results about intervals can be found here.

Technical Note

The $\LaTeX$ code for \(\hointl {a} {b}\) is \hointl {a} {b} .

This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation.

The name is derived from half-open interval on the left.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $39$. Order Topology: $1$