Definition:Accumulation Point

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Definition

Let $\paren { S, \tau }$ be a topological space.

Let $A \subseteq S$.


Accumulation Point of Sequence

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.

Let $x \in S$.

Suppose that:

$\forall U \in \tau: x \in U \implies \set {n \in \N: x_n \in U}$ is infinite


Then $x$ is an accumulation point of $\sequence {x_n}$.


Accumulation Point of Set

Let $x \in S$.


Then $x$ is an accumulation point of $A$ if and only if:

$x \in \cl {A \setminus \set x}$

where $\operatorname{cl}$ denotes the (topological) closure of a set.


Also see

  • Results about Accumulation Points can be found here.