# Limit point

## Definition

### Common form

For a Topological space $(X,\mathcal{J})$, $x\in X$ is a limit point of $A$ if every neighbourhood of $x$ has a non-empty intersection with $A$ that contains some point other than $x$ itself.

### Equivalent form

$x$ is a limit point of $A$ if $x\in\text{Closure}(A-\{x\})$ (you can read about closure here)

TODO: Prove these are the same

## Other names

• Accumilation point

## Examples

### $0$ is a limit point of $(0,1)$

#### Proof using first definition

Is is clear we are talking about the Euclidian metric