Definition:Limit Point
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:
- $S_\xi := \set {x: x \in S, x \ne \xi}$
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 $M = \struct {S, d}$ be a metric space.
Let $\tau$ be the topology induced by the metric $d$.
Let $A \subseteq S$ be a subset of $S$.
Let $\alpha \in S$.
Definition 1
$\alpha$ is a limit point of $A$ if and only if every deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$:
- $\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$
that is:
- $\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$
Note that $\alpha$ does not have to be an element of $A$ to be a limit point.
Definition 2
$\alpha$ is a limit point of $A$ if and only if there is a sequence $\sequence{\alpha_n}$ in $A \setminus \set \alpha$ such that $\sequence{\alpha_n}$ converges to $\alpha$, that is, $\alpha$ is a limit of the sequence $\sequence{\alpha_n}$ in $S$.
Definition 3
$\alpha$ is a limit point of $A$ if and only if $\alpha$ is a limit point in the topological space $\struct{S, \tau}$.
Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $Y \subseteq X$ be a subset of $X$.
Limit Point of Set
Let $\alpha \in X$.
Then $\alpha$ is a limit point of $Y$ if and only if every deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $Y$:
- $\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap Y \ne \O$
that is:
- $\forall \epsilon \in \R_{>0}: \set {x \in Y: 0 < \norm {x - \alpha} < \epsilon} \ne \O$
Note that $\alpha$ does not have to be an element of $A$ to be a limit point.
Limit Point of Sequence
Let $L \in Y$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $Y \setminus \set L$.
Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.
Then $L$ is a limit of $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity which is usually written:
- $\ds L = \lim_{n \mathop \to \infty} x_n$
Topology
Let $T = \struct {S, \tau}$ be a topological space.
Limit Point of Set
A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:
- $A \cap \paren {U \setminus \set x} \ne \O$
That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.
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 Filter
Let $T = \struct {S, \tau}$ be a topological space.
Let $\FF$ be a filter on $S$.
A point $x \in S$ is called a limit point of $\FF$ if and only if $\FF$ is finer than the neighborhood filter of $x$.
Limit Point of Filter Basis
Let $T = \struct {S, \tau}$ be a topological space.
Let $\FF$ be a filter on the underlying set $S$ of $T$.
Let $\BB$ be a filter basis of $\FF$.
Definition 1
A point $x \in S$ is called a limit point of $\BB$ if and only if $\FF$ converges on $x$.
$\BB$ is likewise said to converge on $x$.
Definition 2
A point $x \in S$ is called a limit point of $\BB$ if and only if every neighborhood of $x$ contains a set of $\BB$.
Also known as
A limit point is also known as:
However, note that an accumulation point is also seen with a subtly different definition from that of a limit point, so be careful.
Examples
End Points of Real Interval
The real number $a$ is a limit point of both the open real interval $\openint a b$ as well as of the closed real interval $\closedint a b$.
It is noted that $a \in \closedint a b$ but $a \notin \openint a b$.
Union of Singleton with Open Real Interval
Let $\R$ be the set of real numbers.
Let $H \subseteq \R$ be the subset of $\R$ defined as:
- $H = \set 0 \cup \openint 1 2$
Then $0$ is not a limit point of $H$.
Real Number is Limit Point of Rational Numbers in Real Numbers
Let $\R$ be the set of real numbers.
Let $\Q$ be the set of rational numbers.
Let $x \in \R$.
Then $x$ is a limit point of $\Q$.
Zero is Limit Point of Integer Reciprocal Space
Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
- $A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Then $0$ is the only limit point of $A$ in $\R$.
Also see
- Results about limit points can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): limit point