View source for Equivalent statements to a set being dense

{{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}}
: See [[Motivation:Dense set]] for the motivation of [[dense set]]. This page describes equivalent conditions to a set being [[dense]].
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==Statements==
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:
# {{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:
# {{M|1=\forall U\in\mathcal{J}[U\ne\emptyset\implies\exists y\in E[y\in U]]}}
# {{M|1=\forall U\in\mathcal{J}\exists y\in E[U\ne\emptyset\implies y\in U]}}
# (Obvious permutations of these)
{{Todo|Show them and be certain myself. I can ''believe'' these are equivalent, but I have not shown it!}}</ref>{{rFAVIDMH}}
#* [[A set is dense if and only if every non-empty open subset contains a point of it]]{{rITTMJML}} - definition in {{rFAVIDMH}}
# The [[closure]] of {{M|E}} is {{M|X}} itself{{rFAVIDMH}}
#* This is the definition we use and the definition given by{{rITTMJML}}.
# {{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} }}))
#* {{M|X-E}} contains no ''[[non-empty]]'' [[open set]] of {{M|X}}{{rFAVIDMH}}
# {{XXX|Symbolic form}}{{rFAVIDMH}}
#* {{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)
{{Todo|Factor these out into their own pages and link to}}
===[[Metric space]] cases===
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}}:
# {{M|1=\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset]}}{{rFAVIDMH}}{{rW2014LNFARS}}
#* Words
#* 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}}
{{Todo|Factor these out into their own pages and link to}}
==Proof of claims==
Dense {{iff}} [[A set is dense if and only if every non-empty open subset contains a point of it]] is done already!
{{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}}}}
===Metric spaces claims===
==Notes==
<references group="Note"/>
==References==
<references/>
{{Theorem Of|Topology|Metric Space}}