Closed set

Definitions

Topology

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

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

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 [math]\exists \delta>0:B_\delta(x)\subset N[/math]

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