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
Clopen set
Results
Closed set less open set is closed
Empty set is closed
Finite sets are closed in Fréchet space
Finite sets are closed in Hausdorff space
Finite sets are closed in metric space
Finite union of closed sets is closed
Intersection of closed sets is closed
Singletons are closed in Polish space
Whole space is closed