(i) | \begin{equation} X \neq \emptyset \end{equation} |

(ii) | $a, b \in X$ are each a D2218: Set element in $X$ |

**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}

Definition D1809

Closed interval

Formulation 1

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

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}

(i) | \begin{equation} X \neq \emptyset \end{equation} |

(ii) | $a, b \in X$ are each a D2218: Set element in $X$ |

Formulation 2

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that

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}

(i) | \begin{equation} X \neq \emptyset \end{equation} |

(ii) | $a, b \in X$ are each a D2218: Set element in $X$ |

Children

▶ | Closed real interval |

▶ | Left-closed interval |

▶ | Open interval |

▶ | Right-closed interval |