# Equivalent statements to a set being dense

**Stub grade: A***

- See Motivation:Dense set for the motivation of dense set. This page describes equivalent conditions to a set being dense.

## Contents

## Statements

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]E\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. Then "[ilmath]E[/ilmath] is dense in [ilmath](X,\mathcal{ J })[/ilmath]" is equivalent to any of the following:

- [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies U\cap E\ne\emptyset][/ilmath]
^{[Note 1]}^{[1]}- A set is dense if and only if every non-empty open subset contains a point of it
^{[2]}- definition in^{[1]}

- A set is dense if and only if every non-empty open subset contains a point of it
- The closure of [ilmath]E[/ilmath] is [ilmath]X[/ilmath] itself
^{[1]}- This is the definition we use and the definition given by
^{[2]}.

- This is the definition we use and the definition given by
- [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies \neg(U\subseteq X-E)][/ilmath]
^{[1]}(I had to use negation/[ilmath]\neg[/ilmath] as \not{\subseteq} doesn't render well ([ilmath]\not{\subseteq} [/ilmath])) - TODO: Symbolic form
^{[1]}- [ilmath]X-E[/ilmath] has no interior points
^{[1]}(i.e: [ilmath]\text{interior}(E)=E^\circ=\emptyset[/ilmath], the interior of [ilmath]E[/ilmath] is empty)

- [ilmath]X-E[/ilmath] has no interior points

TODO: Factor these out into their own pages and link to

### Metric space cases

Suppose [ilmath](X,d)[/ilmath] is a metric space and [ilmath](X,\mathcal{ J })[/ilmath] is the topological space induced by the metric space, then the following are equivalent to an arbitrary subset of [ilmath]X[/ilmath], [ilmath]E\in\mathcal{P}(X)[/ilmath] being dense in [ilmath](X,\mathcal{ J })[/ilmath]:

- [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset][/ilmath]
^{[1]}^{[3]}- Words
- This is obviously the same as: [ilmath]\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)][/ilmath] - definition in
^{[3]}

TODO: Factor these out into their own pages and link to

## Proof of claims

Dense *if and only if* A set is dense if and only if every non-empty open subset contains a point of it is done already!

The message provided is:

^{[1]}

### Metric spaces claims

## Notes

- ↑ In the interests of making the reader aware of caveats of formal logic as well as differences, this is equivalent to:
- [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies\exists y\in E[y\in U]][/ilmath]
- [ilmath]\forall U\in\mathcal{J}\exists y\in E[U\ne\emptyset\implies y\in U][/ilmath]
- (Obvious permutations of these)

TODO: Show them and be certain myself. I can

*believe*these are equivalent, but I have not shown it!

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}^{1.7}^{1.8}Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha - ↑
^{2.0}^{2.1}Introduction to Topological Manifolds - John M. Lee - ↑
^{3.0}^{3.1}Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp