Definition:Limit Point/Topology/Set

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.


Let $A \subseteq S$.


Definition from Open Neighborhood

A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:

$A \cap \left({U \setminus \left\{{x}\right\}}\right) \ne \varnothing$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.


Definition from Closure

A point $x \in S$ is a limit point of $A$ if and only if

$x$ belongs to the closure of $A$ but is not an isolated point of $A$.


Definition from Adherent Point

A point $x \in S$ is a limit point of $A$ if and only if $x$ is an adherent point of $A$ but is not an isolated point of $A$.


Definition from Relative Complement

A point $x \in S$ is a limit point of $A$ if and only if $\left({S \setminus A}\right) \cup \left\{{x}\right\}$ is not a neighborhood of $x$.


Definition from Sequence

A point $x \in S$ is a limit point of $A$ if there is a sequence $\left\langle{x_n}\right\rangle$ in $A$ such that $x$ is a limit point of $\left\langle{x_n}\right\rangle$, considered as sequence in $S$.


Also see

  • Results about limit points can be found here.