ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Subset
▼ Power set
▼ Hyperpower set sequence
▼ Hyperpower set
▼ Hypersubset
▼ Subset algebra
▼ Subset structure
▼ Topological space
▼ Open set
Definition D98
Closed set
Formulation 0
Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
A D11: Set $F \subseteq X$ is closed in $T$ if and only if \begin{equation} X \setminus F \in \mathcal{T} \end{equation}
Children
▶ D2640: Clopen set
Results
▶ R1762: Closed set less open set is closed
▶ R76: Empty set is closed
▶ R2746: Finite sets are closed in Fréchet space
▶ R2747: Finite sets are closed in Hausdorff space
▶ R2748: Finite sets are closed in metric space
▶ R74: Finite union of closed sets is closed
▶ R75: Intersection of closed sets is closed
▶ R4007: Singletons are closed in Polish space
▶ R233: Whole space is closed