Definition:Limit Point/Topology

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Definition

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


Limit Point of Set

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 $x$.


Limit Point of Point

The concept of a limit point can be sharpened to apply to individual points, as follows:


Let $a \in S$.


A point $x \in S, x \ne a$ is a limit point of $a$ if and only if every open neighborhood of $x$ contains $a$.

That is, it is a limit point of the singleton $\set a$.


Limit Point of Sequence

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

Let $\sequence {x_n}$ be a sequence in $A$.

Let $\sequence {x_n}$ converge to a value $\alpha \in A$.


Then $\alpha$ is known as a limit (point) of $\sequence {x_n}$ (as $n$ tends to infinity).


Simple Examples


Also

  • Results about limit points can be found here.