Difference between revisions of "Closed set"

Latest revision as of 15:36, 24 November 2015

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

• Relatively closed
• Open set
• Neighbourhood

Notes

1. ↑ Maurin proves this as an [ilmath]\iff[/ilmath] theorem. However he assumes the space is complete.

References

1. ↑ Introduction to topology - Third Edition - Mendelson
2. ↑ Krzyzstof Maurin - Analysis - Part I: Elements