# 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:

## 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]}

- 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]

## 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