ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
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▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
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▼ Preordering relation
▼ Partial ordering relation
▼ Partially ordered set
Definition D1809
Closed interval
Formulation 1
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) \begin{equation} X \neq \emptyset \end{equation}
(ii) $a, b \in X$ are each a D2218: Set element in $X$
The closed interval in $P$ from $a$ to $b$ is the D11: Set \begin{equation} [a, b] : = \{ x \in X : a \preceq x \preceq b \} \end{equation}
Formulation 2
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) \begin{equation} X \neq \emptyset \end{equation}
(ii) $a, b \in X$ are each a D2218: Set element in $X$
The closed interval in $P$ from $a$ to $b$ is the D11: Set \begin{equation} [a, b] : = \{ x \in X : a \preceq x, x \preceq b \} \end{equation}
Children
▶ D544: Closed real interval
▶ D1811: Left-closed interval
▶ D1810: Open interval
▶ D1812: Right-closed interval