ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Partially ordered set
Closed interval
Definition D1811
Left-closed interval
Formulation 0
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 $X$
(iii) ${\prec}$ is the D5350: Partial ordering relation strictisation of ${\preceq}$ on $X$
The left-closed interval in $P$ from $a$ to $b$ is the D11: Set \begin{equation} [a, b) : = \{ x \in X : a \preceq x \prec b \} \end{equation}
Left-closed real interval