▾ 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
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}
Dual definition
» Open set
Child definitions
» D2640: Clopen set
Results
» R76: Empty set is closed
» R74: Finite union of closed sets is closed
» R233: Whole space is closed
» R1762: Closed set less open set is closed
» R2747: Finite sets are closed in Hausdorff space
» R2746: Finite sets are closed in Fréchet space
» R2748: Finite sets are closed in metric space
» R4007: Singletons are closed in Polish space
» R75: Intersection of closed sets is closed