# Equivalent statements to a set being dense

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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.

## 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:

1. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies U\cap E\ne\emptyset][/ilmath][Note 1][1]
2. The closure of [ilmath]E[/ilmath] is [ilmath]X[/ilmath] itself[1]
• This is the definition we use and the definition given by[2].
3. [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]))
4. 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)

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]:

1. [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

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Would be good to at least post a picture of the work, routine and proofs are abundently available, see page 74 in[1]

## Notes

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

TODO: Show them and be certain myself. I can believe these are equivalent, but I have not shown it!