Dense

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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]
   • There are some equivalent conditions
     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
        • 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:

• [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:

• [ilmath]\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies y\in U][/ilmath], which is obviously equivalent to: [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset \implies A\cap U\ne\emptyset][/ilmath] (see Claim 1 below)
  • In words:
    • A set is dense if and only if every non-empty open subset contains a point of it
    • Which can be found in "equivalent statements to a set being dense".

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:

• [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:} (see Claim 1)

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:

• The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set

Grade: C

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

Grade: C

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

See also

• Equivalent statements to a set being dense
  • A set is dense if and only if every non-empty open subset contains a point of it