Definition:Limit Point

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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 $S$ if and only if $\xi$ is at zero distance from $S_\xi$.


Complex Analysis

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

Let $z_0 \in \C$.

Let $\map {N_\epsilon} {z_0}$ 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 $S$ if and only if every deleted $\epsilon$-neighborhood $\map {N_\epsilon} {z_0} \setminus \set {z_0}$ of $z_0$ contains a point in $S$:

$\forall \epsilon \in \R_{>0}: \paren {\map {N_\epsilon} {z_0} \setminus \set {z_0} } \cap S \ne \O$

that is:

$\forall \epsilon \in \R_{>0}: \set {z \in S: 0 < \cmod {z - z_0} < \epsilon} \ne \O$


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 $B_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\}$ of $\alpha$ contains a point in $A$:

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

that is:

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

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


Topology

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 $T = \left({S, \tau}\right)$ 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).


Limit Point of Filter

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

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


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


Limit Point of Filter Basis

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

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


Definition 1

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

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


Definition 2

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


Also known as

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.