ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Binary endorelation ▼ Preordering relation ▼ Partial ordering relation ▼ Partially ordered set ▼ Closed interval
Definition D1810
Open interval

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
 (i) $$X \neq \emptyset$$ (ii) $a, b \in X$ are each a D2218: Set element in $X$ (iii) ${\prec}$ is the D5350: Partial ordering relation strictisation of ${\preceq}$ on $X$
The open interval in $P$ from $a$ to $b$ is the D11: Set $$(a, b) : = \{ x \in X : a \prec x \prec b \}$$

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
 (i) $$X \neq \emptyset$$ (ii) $a, b \in X$ are each a D2218: Set element in $X$ (iii) ${\prec}$ is the D5350: Partial ordering relation strictisation of ${\preceq}$ on $X$
The open interval in $P$ from $a$ to $b$ is the D11: Set $$(a, b) : = \{ x \in X : a \prec x, x \prec b \}$$