Closed set
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Definition
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^{[1]}^{[2]}.
Metric space
- Note: as every metric space is also a topological space it is still true that a set is closed if its complement is open.
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 [math]\exists \delta>0:B_\delta(x)\subset N[/math], then:
- 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]
TODO: This proof could be nonsense
See also
Notes
- ↑ Maurin proves this as an [ilmath]\iff[/ilmath] theorem. However he assumes the space is complete.