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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/>
'''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 {{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.
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>
'''''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==
===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)])}}}}
===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==
* [[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]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology|Metric Space|Functional Analysis}}