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
Definition D1945
Closure point
Formulation 0
Let $T = (X, \mathcal{T})$ be a D1106: Topological space such that
(i) $E \subseteq X$ is a D78: Subset
A D2218: Set element $x \in X$ is a closure point of $E$ in $T$ if and only if \begin{equation} \forall \, U \in \mathcal{T} \, (x \in U \quad \implies \quad E \cap U \neq \emptyset) \end{equation}
Formulation 1
Let $T = (X, \mathcal{T})$ be a D1106: Topological space such that
(i) $E \subseteq X$ is a D78: Subset
A D2218: Set element $x \in X$ is a closure point of $E$ in $T$ if and only if \begin{equation} \forall \, U \in \mathcal{T} \, (x \in U \quad \implies \quad \exists \, y \in X : y \in E \cap U) \end{equation}
Also known as
Proximate point, Contact point, Adherent point
Children
▶ D92: Limit point
▶ D88: Set closure