# Closed set

(Redirected from Closed sets)

## Definition

A closed set in a topological space $(X,\mathcal{J})$ is a set $A$ where $X-A$ is open[1][2].

### Metric space

A subset [ilmath]A[/ilmath] of the metric space [ilmath](X,d)[/ilmath] is closed if it contains all of its limit points[Note 1]

For convenience only: recall [ilmath]x[/ilmath] is a limit point if every neighbourhood of [ilmath]x[/ilmath] contains points of [ilmath]A[/ilmath] other than [ilmath]x[/ilmath] itself.

## Example

[ilmath](0,1)[/ilmath] is not closed, as take the point [ilmath]0[/ilmath].

#### Proof

Let [ilmath]N[/ilmath] be any neighbourhood of [ilmath]x[/ilmath], then $\exists \delta>0:B_\delta(x)\subset N$, then:

• Take $y=\text{Max}\left(\frac{1}{2}\delta,\frac{1}{2}\right)$, then $y\in(0,1)$ and $y\in N$ thus [ilmath]0[/ilmath] is certainly a limit point, but [ilmath]0\notin(0,1)[/ilmath]

TODO: This proof could be nonsense