Definition:Limit Point

Topology

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 $S$.

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 called a limit point of $a$ if every open set $U \in \vartheta$ such that $x \in U$ contains $a$.

It can be seen that this is the same definition as for the definition of a limit point of a set, by requiring that the limit point for a point $a$ is defined as the limit point of the set $\left\{{a}\right\}$.

Limit Point of Sequence

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

Let $A \subseteq S$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $\alpha \in A$.

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

Metric Space

Let $M = \left({S, d}\right)$ be a metric space.

Let $A \subseteq S$ be a subset of $S$.

Let $\alpha \in S$.

Then $\alpha$ is a limit point of $A$ iff every deleted $\epsilon$-neighborhood $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\}$ of $\alpha$ contains a point in $A$ other than $\alpha$:

$\forall \epsilon \in \R, \epsilon > 0: N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} \cap A \ne \varnothing$

that is:

$\forall \epsilon \in \R, \epsilon > 0: \left\{{x \in A: 0 < d \left({x, \alpha}\right) < \epsilon}\right\} \ne \varnothing$

Note that $\alpha$ does not have to be an element of $A$ to be a limit point.

(Informally speaking, $\alpha$ is a limit point of $A$ if there are points in $A$ that are different from $\alpha$ but arbitrarily close to it.)

Complex Analysis

Let $S \subseteq \C$ be a subset of the set of complex numbers.

Let $z_0 \in \C$.

Let $N_\epsilon \left({z_0}\right)$ be the $\epsilon$-neighborhood of $z_0$ for a given $\epsilon \in \R$ such that $\epsilon > 0$.

Then $z_0$ is a limit point of $S$ iff every deleted $\epsilon$-neighborhood $N_\epsilon \left({z_0}\right) \setminus \left\{{z_0}\right\}$ of $z_0$ contains a point in $S$ other than $z_0$:

$\forall \epsilon \in \R, \epsilon > 0: \left({N_\epsilon \left({z_0}\right) \setminus \left\{{z_0}\right\}}\right) \cap S \ne \varnothing$

that is:

$\forall \epsilon \in \R, \epsilon > 0: \left\{{z \in S: 0 < \left|{z - z_0}\right| < \epsilon}\right\} \ne \varnothing$

Note that $z_0$ does not have to be an element of $S$ to be a limit point, although it may well be.

Informally, there are points in $S$ which are arbitrarily close to it.

Real Analysis

Let $S \subseteq \R$ be a subset of the real numbers.

Let $\xi \in \R$ and let $S_\xi$ be the set defined as:

$S_\xi := \left\{{x: x \in S, x \ne \xi}\right\}$

Then $\xi$ is a limit point of $S$ iff $\xi$ is at zero distance from $S_\xi$.

Limit Point of Filter

Let $\mathcal F$ be a filter on a set $S$.

A point $x \in S$ is called a limit point of $\mathcal F$ if:

$\displaystyle x \in \bigcap \left\{{\complement_S \left({V}\right) : V \in \mathcal F}\right\}$

where $\complement_S \left({V}\right)$ is the complement of $V$ relative to $S$.

$\mathcal F$ is said to converge on $x$.

Alternative Definition

Let $\mathcal F$ be a filter on $X$.

A point $x \in X$ is called a limit point of $\mathcal F$ if $\mathcal F$ is finer than the neighborhood filter of $x$.

Limit Point of Filter Base

Let $\mathcal F$ be a filter on a set $S$.

Let $\mathcal B$ be a filter basis of $\mathcal F$.

A point $x \in S$ is called a limit point of $\mathcal B$ if $\mathcal F$ converges on $x$.

$\mathcal B$ is likewise said to converge on $x$.

Alternative Definition

A point $x \in S$ is called a limit point of $\mathcal B$ iff every neighborhood of $x$ contains a set of $\mathcal B$.

Alternative names

A limit point is also known as a cluster point, or a point of accumulation.

However, note that an accumulation point is also seen with a subtly different definition from that of a limit point, so be careful.