Definition:Limit Point/Topology
Definition
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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 $\set a$.
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 $\R$.
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- Every point of $\R$ is a limit point of $\R$ in the usual (Euclidean) topology.
- The set $\Z$ has no limit points in the usual (Euclidean) topology of $\R$.
Also see
- Results about limit points can be found here.