Definition:Interval/Ordered Set

Definition

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

Let $a, b \in S$.

The intervals between $a$ and $b$ are defined as follows:

Open Interval

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 $b$.

Left Half-Open Interval

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 $b$.

Right Half-Open Interval

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 $b$.

Closed Interval

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

$\closedint a b := a^\succcurlyeq \cap b^\preccurlyeq = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }$

where:

$a^\succcurlyeq$ denotes the upper closure of $a$

$b^\preccurlyeq$ denotes the lower closure of $b$.

Endpoint

The elements $a, b \in S$ are known as the endpoints (or end points) of the interval.

$a$ is sometimes called the left hand endpoint and $b$ the right hand end point of the interval.

Wirth Interval Notation

The notation used on this site to denote an interval of an ordered set $\struct {S, \preccurlyeq}$ is a fairly recent innovation, and was introduced by Niklaus Emil Wirth:

\(\ds \openint a b\)

\(:=\)

\(\ds \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }\)

Open Interval

\(\ds \hointr a b\)

\(:=\)

\(\ds \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }\)

Right Half-Open Interval

\(\ds \hointl a b\)

\(:=\)

\(\ds \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }\)

Left Half-Open Interval

\(\ds \closedint a b\)

\(:=\)

\(\ds \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }\)

Closed Interval

Also see

• Definition:Ray (Order Theory)

• Results about intervals can be found here.

Sources

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