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
Closure point
Definition D92
Limit 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 limit 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 \setminus \{ x \})) \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 limit 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 \text{ and } x \neq y) \end{equation}