Definition:Limit Point of Set

Definition

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

Limit Point of Set

Let $A \subseteq S$.

Definition from Open Set

A point $x \in S$ is called a limit point of $A$ if every open set $U \in \vartheta$ such that $x \in U$ contains some point of $A$ other than $x$.

Definition from Closure

$x$ is called a limit point of $A$ if $x$ belongs to the closure of $A$ but is not an isolated point of $A$.

Definition from Adherent Point

$x$ is called a limit point of $A$ if $x$ is an adherent point of $A$ but is not an isolated point of $A$.

Definition from Sequence

$x$ is called 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 $S$.

Definition from Neighborhood

Some sources define a point $x \in S$ to be a limit point of $A$ if every neighborhood $U$ of $x$ satisfies $A \cap \left({U \setminus \left\{{x}\right\}}\right) \ne \varnothing$.

This definition is the same as the definition in terms of a an open set if the definition of neighborhood is the one which insists that the neighborhood must be open in $T$.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Limit Points