Difference between revisions of "Compactness"
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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. | 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}} | {{Begin Proof}} | ||
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'''{{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''' | '''{{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}}) | :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}}) | ||
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:Then <math>\{A_{\alpha_i}\}^n_{i=1}</math> is a sub-collection of <math>\mathcal{A}</math> that covers <math>Y</math>. | :Then <math>\{A_{\alpha_i}\}^n_{i=1}</math> is a sub-collection of <math>\mathcal{A}</math> that covers <math>Y</math>. | ||
| − | '''Proof of details''' | + | '''Proof of details''' - If the reader is not convinced that {{M|1=Y\subseteq\cup_{i=1}^nA_n}} we prove that here. |
:As [[The intersection of sets is a subset of each set]] and <math>\cup^n_{i=1}(A_{\alpha_i}\cap Y)=Y</math> we see <br /> | :As [[The intersection of sets is a subset of each set]] and <math>\cup^n_{i=1}(A_{\alpha_i}\cap Y)=Y</math> we see <br /> | ||
:<math>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</math> <math>\implies x\in A_{\alpha_k}\implies x\in\cup^n_{i=1}A_{\alpha_i}</math><br /> | :<math>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</math> <math>\implies x\in A_{\alpha_k}\implies x\in\cup^n_{i=1}A_{\alpha_i}</math><br /> | ||
:The important part being <math>x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies x\in\cup^n_{i=1}A_{\alpha_i}</math><br /> | :The important part being <math>x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies x\in\cup^n_{i=1}A_{\alpha_i}</math><br /> | ||
| − | :then by the [[Implies and subset relation|implies and subset relation]] we have <math>Y=\cup^n_{i=1}(A_{\alpha_i}\cap Y)\ | + | :then by the [[Implies and subset relation|implies and subset relation]] we have <math>Y=\cup^n_{i=1}(A_{\alpha_i}\cap Y)\subseteq\cup^n_{i=1}A_{\alpha_i}</math> and conclude <math>Y\subseteq\cup^n_{i=1}A_{\alpha_i}</math> |
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Revision as of 17:04, 14 August 2015
See Notes:Compactness and sequences - I think there's a different definition for metric spaces, I have not seen a proof that the metric one [ilmath]\implies[/ilmath] this one
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 [ilmath][0,5]\subset\mathbb{R} [/ilmath]) with open sets. Suppose we have the covering [ilmath]\{(-1,3),(2,6)\} [/ilmath] this is already finite and covers the interval. The corresponding sets in the subspace topology are [ilmath]\{[0,3),(2,5]\} [/ilmath] which are both open in the subspace topology.
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
Definition
- A topological space is compact[1] if every 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:
- [ilmath]\mathcal{A}=\{A_\alpha\}_{\alpha\in I}[/ilmath] such that each [ilmath]A_\alpha[/ilmath] is open in [ilmath]X[/ilmath] and
- [math]X=\bigcup_{\alpha\in I}A_\alpha[/math][Note 1]
The following is true:
- [ilmath]\exists \{i_1,\cdots,i_n\}\subset I[/ilmath] such that [math]X=\bigcup_{\alpha\in\{i_1,\cdots,i_n\} }A_\alpha[/math]
Then [ilmath]X[/ilmath] is compact[1]
Lemma for a set being compact
Take a set [math]Y\subset X[/math] in a topological space [math](X,\mathcal{J})[/math]. Then to say:
- [math]Y[/math] is compact
Means [math]Y[/math] satisfies the definition of compactness when considered as a subspace of [math](X,\mathcal{J})[/math]
Theorem: A set [ilmath]Y\subseteq X[/ilmath] is a compact in [ilmath](X,\mathcal{J})[/ilmath] if and only if every covering of [ilmath]Y[/ilmath] by sets open in [ilmath]X[/ilmath] contains a finite subcovering.
[ilmath](Y,\mathcal{J}_\text{subspace})[/ilmath] is compact [ilmath]\implies[/ilmath] every covering of [ilmath]Y[/ilmath] by sets open in [ilmath]X[/ilmath] 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 [ilmath]Y\subseteq\cup_{\alpha\in I}A_\alpha[/ilmath])
- Then the collection [math]\{A_\alpha\cap Y|\alpha\in I\}[/math] is a covering of [math]Y[/math] by sets open in [math]Y[/math] (by definition of being a subspace)
- By hypothesis [math]Y[/math] is compact, hence a finite sub-collection [math]\{A_{\alpha_i}\cap Y\}^n_{i=1}[/math] covers [math]Y[/math] (as to be compact every open cover must have a finite subcover)
- Then [math]\{A_{\alpha_i}\}^n_{i=1}[/math] is a sub-collection of [math]\mathcal{A}[/math] that covers [math]Y[/math].
Proof of details - If the reader is not convinced that [ilmath]Y\subseteq\cup_{i=1}^nA_n[/ilmath] we prove that here.
- As The intersection of sets is a subset of each set and [math]\cup^n_{i=1}(A_{\alpha_i}\cap Y)=Y[/math] we see
- [math]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[/math] [math]\implies x\in A_{\alpha_k}\implies x\in\cup^n_{i=1}A_{\alpha_i}[/math]
- The important part being [math]x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies x\in\cup^n_{i=1}A_{\alpha_i}[/math]
- then by the implies and subset relation we have [math]Y=\cup^n_{i=1}(A_{\alpha_i}\cap Y)\subseteq\cup^n_{i=1}A_{\alpha_i}[/math] and conclude [math]Y\subseteq\cup^n_{i=1}A_{\alpha_i}[/math]
[ilmath](Y,\mathcal{J}_\text{subspace})[/ilmath] is compact [math]\impliedby[/math] every covering of [ilmath]Y[/ilmath] by sets open in [ilmath]X[/ilmath] contains a finite subcovering
- Suppose that every covering of [math]Y[/math] by sets open in [math]X[/math] contains a finite subcollection covering [math]Y[/math]. We need to show [math]Y[/math] is compact.
- Suppose we have a covering, [math]\mathcal{A}'=\{A'_\alpha\}_{\alpha\in I}[/math] of [math]Y[/math] by sets open in [math]Y[/math]
- For each [math]\alpha[/math] choose an open set [math]A_\alpha[/math] open in [math]X[/math] such that: [math]A'_\alpha=A_\alpha\cap Y[/math]
- Then the collection [math]\mathcal{A}=\{A_\alpha\}_{\alpha\in I}[/math] covers [math]Y[/math]
- By hypothesis we have a finite sub-collection from [ilmath]\mathcal{A} [/ilmath] of things open in [math]X[/math] that cover [math]Y[/math]
- Thus the corresponding finite subcollection of [math]\mathcal{A}'[/math] covers [math]Y[/math]
See also
Notes
- ↑ Note that we actually have [ilmath]X\subseteq\bigcup_{\alpha\in I}A_\alpha[/ilmath] but as topologies are closed under arbitrary union and contain the set the open sets are subsets of we cannot "exceed [ilmath]X[/ilmath]", so we must have [ilmath]X=\bigcup_{\alpha\in I}A_\alpha[/ilmath]
