Difference between revisions of "Compactness"

Revision as of 03:48, 22 June 2015

Not to be confused with Sequential compactness

There are two views here.

Compactness is a topological property and we cannot say a set is compact, we say it is compact and implicitly consider it with the subspace topology
We can say "sure that set is compact".

The difference comes into play when we cover a set (take the interval $[0,5]\subset\mathbb{R}$ ) with open sets. Suppose we have the covering $\{(-1,3),(2,6)\}$ this is already finite and covers the interval. The corresponding sets in the subspace topology are $\{[0,3),(2,5]\}$ which are both open in the subspace topology.

Definition

A topological space is compact^[1] if every open cover of $X$ contains a finite sub-covering that also covers $X$ .

That is to say that given an arbitrary collection of sets:

$\mathcal{A}=\{A_\alpha\}_{\alpha\in I}$ such that each $A_\alpha$ is open in $X$ and
$X=\bigcup_{\alpha\in I}A_\alpha$ ^{[Note 1]}

The following is true:

$\exists \{i_1,\cdots,i_n\}\subset I$ such that $X=\bigcup_{\alpha\in\{i_1,\cdots,i_n\} }A_\alpha$

Then $X$ is compact^[1]

Lemma for a set being compact

Take a set

$Y\subset X$ in a topological space

$(X,\mathcal{J})$ . Then to say:

$Y$ is compact

Means

$Y$ satisfies the definition of compactness when considered as a subspace of

$(X,\mathcal{J})$

[Expand]
Theorem: A set $Y\subseteq X$ is a compact in $(X,\mathcal{J})$ if and only if every covering of $Y$ by sets open in $X$ contains a finite subcovering.

TODO: Redo this proof - it is not very well written

$(Y,\mathcal{J}_\text{subspace})$ is compact $\implies$ every covering of $Y$ by sets open in $X$ contains a finite subcovering

Suppose that the space

$(Y,\mathcal{J}_\text{subspace})$ is compact and that

$\mathcal{A}=\{A_\alpha\}_{\alpha\in I}$ (where each

$A_\alpha\in\mathcal{J}$ - that is each set is open in

$X$ ) is an open covering (which is to say

$Y\subseteq\cup_{\alpha\in I}A_\alpha$ )

Then the collection

$\{A_\alpha\cap Y|\alpha\in I\}$ is a covering of

$Y$ by sets open in

$Y$ (by definition of being a subspace)

By hypothesis

$Y$ is compact, hence a finite sub-collection

$\{A_{\alpha_i}\cap Y\}^n_{i=1}$ covers

$Y$ (as to be compact every open cover must have a finite subcover)

Then

$\{A_{\alpha_i}\}^n_{i=1}$ is a sub-collection of

$\mathcal{A}$ that covers

$Y$ .

Proof of details

As The intersection of sets is a subset of each set and

$\cup^n_{i=1}(A_{\alpha_i}\cap Y)=Y$ we see

$x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies\exists k\in\mathbb{N}\text{ with }1\le k\le n:x\in A_{\alpha_k}\cap Y$

$\implies x\in A_{\alpha_k}\implies x\in\cup^n_{i=1}A_{\alpha_i}$

The important part being

$x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies x\in\cup^n_{i=1}A_{\alpha_i}$

then by the implies and subset relation we have

$Y=\cup^n_{i=1}(A_{\alpha_i}\cap Y)\subset\cup^n_{i=1}A_{\alpha_i}$ and conclude

$Y\subset\cup^n_{i=1}A_{\alpha_i}$

Warning: this next bit looks funny - do not count on!

Lastly, as

$\mathcal{A}$ was a covering

$\cup_{\alpha\in I}A_\alpha=Y$ .

It is clear that

$x\in\cup^n_{i=1}A_{\alpha_i}\implies x\in\cup_{\alpha\in I}A_\alpha$ so again implies and subset relation we have:

$\cup^n_{i=1}A_{\alpha_i}\subset\cup_{\alpha\in I}A_\alpha=Y$ thus concluding

$\cup^n_{i=1}A_{\alpha_i}\subset Y$

Combining

$Y\subset\cup^n_{i=1}A_{\alpha_i}$ and

$\cup^n_{i=1}A_{\alpha_i}\subset Y$ we see

$\cup^n_{i=1}A_{\alpha_i}=Y$

Thus

$\{A_{\alpha_i}\}^n_{i=1}$ is a finite covering of

$Y$ consisting of open sets from

$X$

End of warning - I've left this here because I must have put it in for a reason!

TODO: What was I hoping to do here?

$(Y,\mathcal{J}_\text{subspace})$ is compact every covering of $Y$ by sets open in $X$ contains a finite subcovering

Suppose that every covering of

$Y$ by sets open in

$X$ contains a finite subcollection covering

$Y$ . We need to show

$Y$ is compact.

Suppose we have a covering,

$\mathcal{A}'=\{A'_\alpha\}_{\alpha\in I}$ of

$Y$ by sets open in

$Y$

For each

$\alpha$ choose an open set

$A_\alpha$ open in

$X$ such that:

$A'_\alpha=A_\alpha\cap Y$

Then the collection

$\mathcal{A}=\{A_\alpha\}_{\alpha\in I}$ covers

$Y$

By hypothesis we have a finite sub-collection from

$\mathcal{A}$ of things open in

$X$ that cover

$Y$

Thus the corresponding finite subcollection of

$\mathcal{A}'$ covers

$Y$

Notes

Jump up ↑ Note that we actually have $X\subseteq\bigcup_{\alpha\in I}A_\alpha$ but as topologies are closed under arbitrary union and contain the set the open sets are subsets of we cannot "exceed $X$ ", so we must have $X=\bigcup_{\alpha\in I}A_\alpha$

References

↑ ^{Jump up to: 1.0} ^1.1 Topology - James R. Munkres - Second Edition

[2] Jump up ↑ Note that we actually have $X\subseteq\bigcup_{\alpha\in I}A_\alpha$ but as topologies are closed under arbitrary union and contain the set the open sets are subsets of we cannot "exceed $X$ ", so we must have $X=\bigcup_{\alpha\in I}A_\alpha$

[Topology-1] {Jump up to: 1.0} ^1.1 Topology - James R. Munkres - Second Edition

[1]

[Note 1]

@@ Line 10: / Line 10: @@
 ==Definition==
-A [[Topological space|topological space]] is compact if every [[Covering|open cover]] (often denoted <math>\mathcal{A}</math>) of <math>X</math> contains a finite sub-collection that also covers <math>X</math>
+*A [[Topological space|topological space]] is compact<ref name="Topology">Topology - James R. Munkres - Second Edition</ref> if every [[Covering|open cover]] of <math>X</math> contains a finite sub-covering that also covers <math>X</math>.
+That is to say that given an arbitrary collection of sets:
+* {{M|1=\mathcal{A}=\{A_\alpha\}_{\alpha\in I} }} such that each {{M|A_\alpha}} is [[Open set|open]] in {{M|X}} and
+* {{MM|1=X=\bigcup_{\alpha\in I}A_\alpha}}<ref group="Note">Note that we actually have {{M|X\subseteq\bigcup_{\alpha\in I}A_\alpha}} but as topologies are closed under arbitrary union and contain the set the open sets are subsets of we cannot "exceed {{M|X}}", so we must have {{M|1=X=\bigcup_{\alpha\in I}A_\alpha}}</ref>
+The following is true:
+* {{M|1=\exists \{i_1,\cdots,i_n\}\subset I}} such that {{MM|1=X=\bigcup_{\alpha\in\{i_1,\cdots,i_n\} }A_\alpha}}
+Then {{M|X}} is ''compact''<ref name="Topology"/>
 ==Lemma for a set being compact==
-Take a set <math>Y\subset X</math> in a [[Topological space|topological space]] <math>(X,\mathcal{J})</math>.
+Take a set <math>Y\subset X</math> in a [[Topological space|topological space]] <math>(X,\mathcal{J})</math>. Then to say:
+* <math>Y</math> is compact
-To say <math>Y</math> is compact is for <math>Y</math> to be compact when considered as a [[Subspace topology|subspace]] of <math>(X,\mathcal{J})</math>
+Means <math>Y</math> satisfies the definition of compactness when considered as a [[Subspace topology|subspace]] of <math>(X,\mathcal{J})</math>
-That is to say that <math>Y</math> is compact if and only if every covering of <math>Y</math> by sets open in <math>X</math> contains a finite subcovering covering <math>Y</math>
 {{Begin Theorem}}
-Theorem: A set {{M|Y\subseteq X}} is a compact space (considered with the subspace topology) of {{M|(X,\mathcal{J})}} ''if and only if'' every covering of {{M|Y}} by sets open in {{M|X}} contains a finite subcovering.
+Theorem: A set {{M|Y\subseteq X}} is a compact in {{M|(X,\mathcal{J})}} ''if and only if'' every covering of {{M|Y}} by sets open in {{M|X}} contains a finite subcovering.
 {{Begin Proof}}
+{{Todo|Redo this proof - it is not very well written}}
 '''{{M|(Y,\mathcal{J}_\text{subspace})}} is compact {{M|\implies}} every covering of {{M|Y}} by sets open in {{M|X}} contains a finite subcovering'''
 :Suppose that the space <math>(Y,\mathcal{J}_\text{subspace})</math> is compact and that <math>\mathcal{A}=\{A_\alpha\}_{\alpha\in I}</math> (where each <math>A_\alpha\in\mathcal{J}</math> - that is each set is open in <math>X</math>) is an open covering (which is to say {{M|Y\subseteq\cup_{\alpha\in I}A_\alpha}})
@@ Line 64: / Line 68: @@
 {{End Proof}}
 {{End Theorem}}
+==See also==
+* [[Subspace topology]]
+==Notes==
+<references group="Note"/>
+==References==
+<references/>
 {{Definition|Topology}}

Difference between revisions of "Compactness"

Revision as of 03:48, 22 June 2015

Contents

Definition

Lemma for a set being compact

See also

Notes

References

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