Definition:Interval/Ordered Set/Right Half-Open

Definition

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

Let $a, b \in S$.

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 $b$.

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 right half-open interval is also called:

a half-open interval on the right

a left half-closed interval

a half-closed interval on the left.

Also see

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

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

• Results about intervals can be found here.

Technical Note

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

Sources

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