A set is dense if and only if every non-empty open subset contains a point of it

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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:

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

  1. I was tempted to write:
    • [ilmath]\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies a\in U][/ilmath]
    However this might not be the same! (They actually are but there's some formality to be observed here, we must deal with both the cases of [ilmath]U=\emptyset[/ilmath] and [ilmath]A=\emptyset[/ilmath] I will deal with these in the future

References

  1. Introduction to Topological Manifolds - John M. Lee