Difference between revisions of "Closed set"
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==Definitions== | ==Definitions== | ||
===Topology=== | ===Topology=== | ||
| − | 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<ref>Introduction to topology - Third Edition - Mendelson</ref> 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]] | 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. | 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. | ||
| − | + | ==Example== | |
{{M|(0,1)}} is not closed, as take the point {{M|0}}. | {{M|(0,1)}} is not closed, as take the point {{M|0}}. | ||
====Proof==== | ====Proof==== | ||
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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)}} | 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)}} | ||
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| + | ==See also== | ||
| + | * [[Relatively closed]] | ||
| + | * [[Open set]] | ||
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| + | ==References== | ||
| + | <references/> | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 18:36, 19 April 2015
Contents
[hide]Definitions
Topology
A closed set[1] in a topological space (X,J) is a set A where X−A is open.
Metric space
A subset A of the metric space (X,d) is closed if it contains all of its limit points
For convenience only: recall x is a limit point if every neighbourhood of x contains points of A other than x itself.
Example
(0,1) is not closed, as take the point 0.
Proof
Let N be any neighbourhood of x, then ∃δ>0:Bδ(x)⊂N
Take y=Max(12δ,12), then y∈(0,1) and y∈N thus 0 is certainly a limit point, but 0∉(0,1)
See also
References
- Jump up ↑ Introduction to topology - Third Edition - Mendelson
