Definition:Interval/Ordered Set/Open
< Definition:Interval/Ordered Set(Redirected from Definition:Open Interval)
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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
- 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