Difference between revisions of "Dense"
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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}}: | ||
* {{M|1=\overline{A}=X}} - that is to say that the {{link|closure|set, topology}} of {{M|A}} is the entirety of {{M|X}} itself. | * {{M|1=\overline{A}=X}} - that is to say that the {{link|closure|set, topology}} of {{M|A}} is the entirety of {{M|X}} itself. | ||
| + | Some authors give the following equivalent definition to {{M|A}} being dense{{rFAVIDMH}}: | ||
| + | * [[A set is dense if and only if every non-empty open subset contains a point of it|{{M|1=\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies y\in U]}}]], which is obviously equivalent to: {{M|1=\forall U\in\mathcal{J}[U\ne\emptyset \implies A\cap U\ne\emptyset]}} (see '''''Claim 1''''' below) | ||
| + | ** In words: | ||
| + | *** [[A set is dense if and only if every non-empty open subset contains a point of it]] | ||
| + | *** Which can be found in "''[[equivalent statements to a set being dense]]''". | ||
| + | ===[[Metric spaces]] definition=== | ||
| + | Let {{M|(X,d)}} me a [[metric space]], we say that {{M|E\in\mathcal{P}(X)}} (so {{M|E}} is an arbitrary [[subset of]] {{M|X}}) if{{rFAVIDMH}}: | ||
| + | * {{M|1=\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset]}} - where {{M|B_r(x)}} denotes the [[open ball]] of [[radius]] {{M|r}}, centred at {{M|x}} | ||
| + | ** 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> | ||
| + | ==Proof of claims== | ||
| + | ===Claim 1=== | ||
| + | This is used for both cases, and it should really be factored out into its own page. Eg: | ||
| + | * [[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)])}}}} | ||
==See also== | ==See also== | ||
* [[Equivalent statements to a set being dense]] | * [[Equivalent statements to a set being dense]] | ||
** [[A set is dense if and only if every non-empty open subset contains a point of it]] | ** [[A set is dense if and only if every non-empty open subset contains a point of it]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | {{Definition|Topology|Metric Space}} | + | {{Definition|Topology|Metric Space|Functional Analysis}} |
Revision as of 19:21, 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]
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
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:
This proof has been marked as an page requiring an easy proof
The message provided is:
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
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)
