Difference between revisions of "Closed set"

Revision as of 00:29, 9 March 2015

A closed set in a topological space

$(X,\mathcal{J})$ is a set

$A$ where

$X-A$ is open.

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.

$(0,1)$ is not closed, as take the point $0$ .

Let $N$ be any neighbourhood of $x$ , then

$\exists \delta>0:B_\delta(x)\subset N$

Take

$y=\text{Max}\left(\frac{1}{2}\delta,\frac{1}{2}\right)$ , then

$y\in(0,1)$ and

$y\in N$ thus

$0$ is certainly a limit point, but

$0\notin(0,1)$

@@ Line 5: / Line 5: @@
 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===
-{{Todo}}
+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]]
+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}}.
+====Proof====
+Let {{M|N}} be any [[Open set#Neighbourhood|neighbourhood]] of {{M|x}}, 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 {{M|0}} is certainly a limit point, but {{M|0\notin(0,1)}}
 {{Definition|Topology}}