Difference between revisions of "Closed set"
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A closed set in a [[Topological space|topological space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X-A</math> is open. | A closed set in a [[Topological space|topological space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X-A</math> is open. | ||
===Metric space=== | ===Metric space=== | ||
| − | {{ | + | A subset {{M|A}} of the [[Metric space|metric space]] {{M|(X,d)}} is closed if it contains all of its [[Limit point|limit points]] |
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| + | For convenience only: recall {{M|x}} is a limit point if every [[Open set#Neighbourhood|neighbourhood]] of {{M|x}} contains points of {{M|A}} other than {{M|x}} itself. | ||
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| + | ===Example=== | ||
| + | {{M|(0,1)}} is not closed, as take the point {{M|0}}. | ||
| + | ====Proof==== | ||
| + | Let {{M|N}} be any [[Open set#Neighbourhood|neighbourhood]] of {{M|x}}, then <math>\exists \delta>0:B_\delta(x)\subset N</math> | ||
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| + | 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 {{M|0}} is certainly a limit point, but {{M|0\notin(0,1)}} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 00:29, 9 March 2015
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]
