Definition:Limit Point/Topology

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Definition

Let $T = \struct {S, \tau}$ 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:

$A \cap \paren {U \setminus \set x} \ne \O$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.


More symbolically, a point $x \in S$ is a limit point of $A$ if and only if

$\forall U\in \tau :x\in U \implies A \cap \paren {U \setminus \set x} \ne \O\text{.}$


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 $\alpha \in S$.


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


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

  • Results about limit points can be found here.