[[Sequence]]
# Limit point

A **limit point**[^Häufungspunkt] generalises [[convergence]] to a limit.
Given a topological space $X$,
a point $a \in X$ is called a **limit point** of a sequence $(x_n)_{n=1}^\infty$ iff every (open) [[neighbourhood]] from $a$ contains infinite $a_n$. #m/def/topology 
Note that this does not imply $x_{n} \to a$,
for examples the sequence defined by $a_{n} = (-1)^n$ is not [[Convergence|convergent]] but has limit points $\{ -1, 1 \}$

[^Häufungspunkt]: German _der Häufungspunkt_

## Properties

- [[Limit points are points contained in the closure of every end piece]]
- [[Limit points are limits of convergent subsequences in a first-countable space]]

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