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

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