Definition:Interval/Ordered Set/Open

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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

  • 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