Definition:Interval/Ordered Set/Open

Definition

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

Let $a, b \in S$.

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

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

where:

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

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

Also defined as

Some sources require that $a \preccurlyeq b$ or $a \prec b$.

Also see

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

• Results about intervals can be found here.

Technical Note

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

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $39$. Order Topology: $1$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations