# Dense

(Redirected from Dense set)
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote.

DENSE IS SPRAWLED OVER LIKE 4 PAGES

• I've distilled some of it Equivalent statements to a set being dense there, but I need to .... fix this page up, it's a mess. I should probably move the equivalent definitions to here, as they're like... "easy equivalent" and may well be definitions, not like ... a proposition of equivalence.
• That's a woolly distinction

Anyway, there is work required to fix this up.

SEE: List of topological properties for a smaller and neater list

## Temporary summary

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, and [ilmath](X,d)[/ilmath] be a metric space. Then for an arbitrary subset of [ilmath]X[/ilmath], say [ilmath]A\in\mathcal{P}(X)[/ilmath], we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if:

1. Topological: [ilmath]\forall U\in\mathcal{J}[U\cap A\neq\emptyset][/ilmath][1]
• There are some equivalent conditions[Note 1]
1. [ilmath]\text{Closure}(A)[/ilmath][ilmath]\eq X[/ilmath] (sometimes written: [ilmath]\overline{A}\eq X[/ilmath])
2. [ilmath]X-A[/ilmath] contains no (non-empty) open subsets of [ilmath]X[/ilmath]
• Symbolically: [ilmath]\forall U\in\mathcal{J}[U\nsubseteq X-A][/ilmath] - which is easily seen to be equivalent to: [ilmath]\forall U\in\mathcal{J}\exists p\in U[p\notin X-A][/ilmath]
3. [ilmath]X-A[/ilmath] has no interior points[Note 2]
• Symbolically we may write this as: [ilmath]\forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right][/ilmath]
[ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)][/ilmath]
[ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))][/ilmath] - by the negation of logical and
[ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A][/ilmath] - of course by the implies-subset relation we see [ilmath](A\subseteq B)\iff(\forall a\in A[a\in B])[/ilmath], thus:
[ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big][/ilmath]
2. Metric: [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap A\neq\emptyset[/ilmath]
• There are no equivalent statements at this time.
Notes to editors: (Alec (talk) 04:15, 1 January 2017 (UTC))

Written by: Alec (talk) 04:15, 1 January 2017 (UTC)

I have used the data at List of topological properties to create this, whilst doing so I added a symbolic form for the interior point statement of topological density.

That symbolic form was added to the list.

The rest of the page continues below. It will be refactored soon.

## Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. We say "[ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if[2]:

• [ilmath]\overline{A}=X[/ilmath] - that is to say that the closure of [ilmath]A[/ilmath] is the entirety of [ilmath]X[/ilmath] itself.

Some authors give the following equivalent definition to [ilmath]A[/ilmath] being dense[1]:

### Metric spaces definition

Let [ilmath](X,d)[/ilmath] me a metric space, we say that [ilmath]E\in\mathcal{P}(X)[/ilmath] (so [ilmath]E[/ilmath] is an arbitrary subset of [ilmath]X[/ilmath]) if[1]:

• [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset][/ilmath] - where [ilmath]B_r(x)[/ilmath] denotes the open ball of radius [ilmath]r[/ilmath], centred at [ilmath]x[/ilmath]
• In words: Every open ball at every point overlaps with [ilmath]E[/ilmath]. (i.e: every open ball at every point contains at least 1 point in common with [ilmath]E[/ilmath])
• This is equivalent to [ilmath]\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)][/ilmath]Found in:[3] (see Claim 1)[Note 3]

Claim 2: for a metric space [ilmath](X,d)[/ilmath] a subset, [ilmath]E\in\mathcal{P}(X)[/ilmath] is dense in the metric sense if and only if it is dense in [ilmath](X,\mathcal{ J })[/ilmath] where [ilmath]J[/ilmath] is the topology induced by the metric [ilmath]d[/ilmath].

## Proof of claims

### Claim 1

This is used for both cases, and it should really be factored out into its own page. Eg:

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:
It is obvious that [ilmath](B_\epsilon(x)\cap E\ne\emptyset)\iff(\exists y\in E[y\in B_\epsilon(x)])[/ilmath]

This proof has been marked as an page requiring an easy proof

### Claim 2

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:
Easy for someone informed of what a metric space and topology is. The claim means showing that if [ilmath](X,\mathcal{ J })[/ilmath] is the topological space induced by a metric space for a metric space [ilmath](X,d)[/ilmath] then [ilmath]E[/ilmath] is dense in [ilmath](X,\mathcal{ J })[/ilmath] if and only if [ilmath]E[/ilmath] is dense in [ilmath](X,d)[/ilmath]

This proof has been marked as an page requiring an easy proof