Definition:Limit Point/Topology/Set

Definition

Let $T = \struct {S, \tau}$ be a topological space.

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 $x$.

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 $\paren {S \setminus A} \cup \set x$ is not a neighborhood of $x$.

Also known as

A limit point is also known as:

a cluster point

an accumulation point.

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

Also see

• Equivalence of Definitions of Limit Point of Set

• Definition:Condensation Point
• Definition:Adherent Point
• Definition:Omega-Accumulation Point

• Relationship between Limit Point Types

• Results about limit points can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 2. (of a set)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 2. (of a set)