Difference between revisions of "Dense"
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** In words: Every [[open ball]] at every point overlaps with {{M|E}}. (i.e: every open ball at every point contains at least 1 point in common with {{M|E}}) | ** In words: Every [[open ball]] at every point overlaps with {{M|E}}. (i.e: every open ball at every point contains at least 1 point in common with {{M|E}}) | ||
** This is equivalent to {{M|1=\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)]}}<sup>Found in:</sup>{{rW2014LNFARS}} (see '''''Claim 1''''')<ref group="Note">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)</ref> | ** This is equivalent to {{M|1=\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)]}}<sup>Found in:</sup>{{rW2014LNFARS}} (see '''''Claim 1''''')<ref group="Note">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)</ref> | ||
| + | '''''Claim 2: ''''' for a [[metric space]] {{M|(X,d)}} a subset, {{M|E\in\mathcal{P}(X)}} is dense in the metric sense {{iff}} it is dense in {{Top.|X|J}} where {{M|J}} is the [[topology induced by the metric]] {{M|d}}. | ||
==Proof of claims== | ==Proof of claims== | ||
===Claim 1=== | ===Claim 1=== | ||
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* [[The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set]] | * [[The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set]] | ||
{{Requires proof|easy=true|grade=C|msg=It is obvious that {{M|1=(B_\epsilon(x)\cap E\ne\emptyset)\iff(\exists y\in E[y\in B_\epsilon(x)])}}}} | {{Requires proof|easy=true|grade=C|msg=It is obvious that {{M|1=(B_\epsilon(x)\cap E\ne\emptyset)\iff(\exists y\in E[y\in B_\epsilon(x)])}}}} | ||
| + | ===Claim 2=== | ||
| + | {{Requires proof|easy=true|grade=C|msg=Easy for someone informed of what a metric space and topology is. The claim means showing that if {{Top.|X|J}} is the [[topological space induced by a metric space]] for a [[metric space]] {{M|(X,d)}} then {{M|E}} is dense in {{Top.|X|J}} {{iff}} {{M|E}} is dense in {{M|(X,d)}}}} | ||
==See also== | ==See also== | ||
* [[Equivalent statements to a set being dense]] | * [[Equivalent statements to a set being dense]] | ||
Revision as of 19:40, 28 October 2016
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Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote
Contents
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]:
- [ilmath]\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies y\in U][/ilmath], which is obviously equivalent to: [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset \implies A\cap U\ne\emptyset][/ilmath] (see Claim 1 below)
- In words:
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:
Grade: C
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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.
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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
- ↑ 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)
