Difference between revisions of "Dense"

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m (Added equivalence claim for metric space definition to their induced topological spaces.)
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{{Stub page|grade=B|msg=Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote}}
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{{Stub page|grade=B|msg=Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote.<br/>
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'''DENSE IS SPRAWLED OVER LIKE 4 PAGES'''
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* I've distilled some of it [[Equivalent statements to a set being dense]] there, but I need to .... fix this page up, it's a mess. I should probably move the equivalent definitions to here, as they're like... "easy equivalent" and may well be definitions, not like ... a proposition of equivalence.
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** That's a woolly distinction
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Anyway, there is work required to fix this up.
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'''SEE: [[List of topological properties]] for a smaller and neater list'''}}
 
==Definition==
 
==Definition==
 
Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. We say "''{{M|A}} is dense in {{M|X}}'' if{{rITTMJML}}:
 
Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. We say "''{{M|A}} is dense in {{M|X}}'' if{{rITTMJML}}:

Revision as of 10:27, 30 December 2016

Stub grade: B
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Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote.

DENSE IS SPRAWLED OVER LIKE 4 PAGES

  • I've distilled some of it Equivalent statements to a set being dense there, but I need to .... fix this page up, it's a mess. I should probably move the equivalent definitions to here, as they're like... "easy equivalent" and may well be definitions, not like ... a proposition of equivalence.
    • That's a woolly distinction

Anyway, there is work required to fix this up.

SEE: List of topological properties for a smaller and neater list

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. We say "[ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if[1]:

  • [ilmath]\overline{A}=X[/ilmath] - that is to say that the closure of [ilmath]A[/ilmath] is the entirety of [ilmath]X[/ilmath] itself.

Some authors give the following equivalent definition to [ilmath]A[/ilmath] being dense[2]:

Metric spaces definition

Let [ilmath](X,d)[/ilmath] me a metric space, we say that [ilmath]E\in\mathcal{P}(X)[/ilmath] (so [ilmath]E[/ilmath] is an arbitrary subset of [ilmath]X[/ilmath]) if[2]:

  • [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset][/ilmath] - where [ilmath]B_r(x)[/ilmath] denotes the open ball of radius [ilmath]r[/ilmath], centred at [ilmath]x[/ilmath]
    • In words: Every open ball at every point overlaps with [ilmath]E[/ilmath]. (i.e: every open ball at every point contains at least 1 point in common with [ilmath]E[/ilmath])
    • This is equivalent to [ilmath]\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)][/ilmath]Found in:[3] (see Claim 1)[Note 1]

Claim 2: for a metric space [ilmath](X,d)[/ilmath] a subset, [ilmath]E\in\mathcal{P}(X)[/ilmath] is dense in the metric sense if and only if it is dense in [ilmath](X,\mathcal{ J })[/ilmath] where [ilmath]J[/ilmath] is the topology induced by the metric [ilmath]d[/ilmath].

Proof of claims

Claim 1

This is used for both cases, and it should really be factored out into its own page. Eg:

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It is obvious that [ilmath](B_\epsilon(x)\cap E\ne\emptyset)\iff(\exists y\in E[y\in B_\epsilon(x)])[/ilmath]

This proof has been marked as an page requiring an easy proof

Claim 2

Grade: C
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:
Easy for someone informed of what a metric space and topology is. The claim means showing that if [ilmath](X,\mathcal{ J })[/ilmath] is the topological space induced by a metric space for a metric space [ilmath](X,d)[/ilmath] then [ilmath]E[/ilmath] is dense in [ilmath](X,\mathcal{ J })[/ilmath] if and only if [ilmath]E[/ilmath] is dense in [ilmath](X,d)[/ilmath]

This proof has been marked as an page requiring an easy proof

See also

Notes

  1. This is a trivial restatement of the claim and not worth going on its own page, unless it can be generalised (eg detached from dense-ness)

References

  1. Introduction to Topological Manifolds - John M. Lee
  2. 2.0 2.1 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
  3. Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp