Difference between revisions of "Equivalent statements to a set being dense"

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(Created page with "{{Stub page|grade=A*|msg=Demote to grade C or D once more theorems have been sought out and the page resembles a page rather than its current state}} ==Statements== * A set...")
 
(Hugely fleshed out, added more cases, added metric space section, added references.)
 
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{{Stub page|grade=A*|msg=Demote to grade C or D once more theorems have been sought out and the page resembles a page rather than its current state}}
 
{{Stub page|grade=A*|msg=Demote to grade C or D once more theorems have been sought out and the page resembles a page rather than its current state}}
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: See [[Motivation:Dense set]] for the motivation of [[dense set]]. This page describes equivalent conditions to a set being [[dense]].
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__TOC__
 
==Statements==
 
==Statements==
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* [[A set is dense if and only if every non-empty open subset contains a point of it]]{{rITTMJML}}
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Let {{Top.|X|J}} be a [[topological space]] and let {{M|E\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then "{{M|E}} is [[dense set|dense]] in {{Top.|X|J}}" is equivalent to any of the following:
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# {{M|1=\forall U\in\mathcal{J}[U\ne\emptyset\implies U\cap E\ne\emptyset]}}<ref group="Note">In the interests of making the reader aware of caveats of formal logic as well as differences, this is equivalent to:
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# {{M|1=\forall U\in\mathcal{J}[U\ne\emptyset\implies\exists y\in E[y\in U]]}}
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# {{M|1=\forall U\in\mathcal{J}\exists y\in E[U\ne\emptyset\implies y\in U]}}
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# (Obvious permutations of these)
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{{Todo|Show them and be certain myself. I can ''believe'' these are equivalent, but I have not shown it!}}</ref>{{rFAVIDMH}}
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#* [[A set is dense if and only if every non-empty open subset contains a point of it]]{{rITTMJML}} - definition in {{rFAVIDMH}}
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# The [[closure]] of {{M|E}} is {{M|X}} itself{{rFAVIDMH}}
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#* This is the definition we use and the definition given by{{rITTMJML}}.
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# {{M|1=\forall U\in\mathcal{J}[U\ne\emptyset\implies \neg(U\subseteq X-E)]}}{{rFAVIDMH}} (I had to use [[negation]]/{{M|\neg}} as {{C|\not{\subseteq}<nowiki/>}} doesn't render well ({{M|\not{\subseteq} }}))
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#* {{M|X-E}} contains no ''[[non-empty]]'' [[open set]] of {{M|X}}{{rFAVIDMH}}
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# {{XXX|Symbolic form}}{{rFAVIDMH}}
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#* {{M|X-E}} has no {{plural|interior point|s}}{{rFAVIDMH}} (i.e: {{M|1=\text{interior}(E)=E^\circ=\emptyset}}, the [[interior]] of {{M|E}} is empty)
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{{Todo|Factor these out into their own pages and link to}}
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===[[Metric space]] cases===
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Suppose {{M|(X,d)}} is a [[metric space]] and {{Top.|X|J}} is the [[topological space induced by the metric space]], then the following are equivalent to an arbitrary [[subset of]] {{M|X}}, {{M|E\in\mathcal{P}(X)}} being dense in {{Top.|X|J}}:
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# {{M|1=\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset]}}{{rFAVIDMH}}{{rW2014LNFARS}}
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#* Words
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#* This is obviously the same as: {{M|1=\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)]}} - definition in {{rW2014LNFARS}}
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{{Todo|Factor these out into their own pages and link to}}
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==Proof of claims==
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Dense {{iff}} [[A set is dense if and only if every non-empty open subset contains a point of it]] is done already!
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{{Requires proof|grade=A|msg=Would be good to at least post a picture of the work, routine and proofs are abundently available, see page 74 in{{rFAVIDMH}}}}
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===Metric spaces claims===
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==Notes==
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<references group="Note"/>
 
==References==
 
==References==
 
<references/>
 
<references/>
 
{{Theorem Of|Topology|Metric Space}}
 
{{Theorem Of|Topology|Metric Space}}

Latest revision as of 20:18, 28 October 2016

Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Demote to grade C or D once more theorems have been sought out and the page resembles a page rather than its current state
See Motivation:Dense set for the motivation of dense set. This page describes equivalent conditions to a set being dense.

Statements

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]E\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. Then "[ilmath]E[/ilmath] is dense in [ilmath](X,\mathcal{ J })[/ilmath]" is equivalent to any of the following:

  1. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies U\cap E\ne\emptyset][/ilmath][Note 1][1]
  2. The closure of [ilmath]E[/ilmath] is [ilmath]X[/ilmath] itself[1]
    • This is the definition we use and the definition given by[2].
  3. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies \neg(U\subseteq X-E)][/ilmath][1] (I had to use negation/[ilmath]\neg[/ilmath] as \not{\subseteq} doesn't render well ([ilmath]\not{\subseteq} [/ilmath]))
  4. TODO: Symbolic form
    [1]
    • [ilmath]X-E[/ilmath] has no interior points[1] (i.e: [ilmath]\text{interior}(E)=E^\circ=\emptyset[/ilmath], the interior of [ilmath]E[/ilmath] is empty)

TODO: Factor these out into their own pages and link to

Metric space cases

Suppose [ilmath](X,d)[/ilmath] is a metric space and [ilmath](X,\mathcal{ J })[/ilmath] is the topological space induced by the metric space, then the following are equivalent to an arbitrary subset of [ilmath]X[/ilmath], [ilmath]E\in\mathcal{P}(X)[/ilmath] being dense in [ilmath](X,\mathcal{ J })[/ilmath]:

  1. [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset][/ilmath][1][3]
    • Words
    • This is obviously the same as: [ilmath]\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)][/ilmath] - definition in [3]

TODO: Factor these out into their own pages and link to

Proof of claims

Dense if and only if A set is dense if and only if every non-empty open subset contains a point of it is done already!

Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Would be good to at least post a picture of the work, routine and proofs are abundently available, see page 74 in[1]

Metric spaces claims

Notes

  1. ↑ In the interests of making the reader aware of caveats of formal logic as well as differences, this is equivalent to:
    1. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies\exists y\in E[y\in U]][/ilmath]
    2. [ilmath]\forall U\in\mathcal{J}\exists y\in E[U\ne\emptyset\implies y\in U][/ilmath]
    3. (Obvious permutations of these)

    TODO: Show them and be certain myself. I can believe these are equivalent, but I have not shown it!

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
  2. ↑ 2.0 2.1 Introduction to Topological Manifolds - John M. Lee
  3. ↑ 3.0 3.1 Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp


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