Difference between revisions of "A set is dense if and only if every non-empty open subset contains a point of it"
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{{Stub page|grade=A*|msg=Demote once this has been checked over. A* because that note isn't dealt with. Also move the statement into its own subpage for transclusion onto [[equivalent statements to a set being dense]]}} | {{Stub page|grade=A*|msg=Demote once this has been checked over. A* because that note isn't dealt with. Also move the statement into its own subpage for transclusion onto [[equivalent statements to a set being dense]]}} | ||
+ | : This is one of a series of theorems: | ||
+ | :* [[Equivalent statements to a set being dense]] | ||
==Statement== | ==Statement== | ||
Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]]. Then we claim{{rITTMJML}}: | Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]]. Then we claim{{rITTMJML}}: |
Latest revision as of 20:29, 28 October 2016
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Demote once this has been checked over. A* because that note isn't dealt with. Also move the statement into its own subpage for transclusion onto equivalent statements to a set being dense
- This is one of a series of theorems:
Contents
Statement
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset. Then we claim[1]:
- [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if and only if
- Symbolically: [ilmath]\forall A\in\mathcal{P}(A)[(\overline{A}=X)\iff(\forall U\in\mathcal{J}[U\ne\emptyset\implies \exists a\in A[a\in U]])[/ilmath][Note 1]
See also
Notes
- ↑ I was tempted to write:
- [ilmath]\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies a\in U][/ilmath]
References