Category:Accumulation Points

This category contains results about Accumulation Points.
Definitions specific to this category can be found in Definitions/Accumulation Points.

Let $\struct {S, \tau}$ 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 $x \in S$.

Then $x \in S$ is an accumulation point of $\sequence {x_n}$ if and only if:

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

Accumulation Point of Set

Let $x \in S$.

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

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

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