Difference between revisions of "Dense"

Latest revision as of 04:15, 1 January 2017

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.

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]:

• [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^[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:

• 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

Claim 2

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

Notes

1. ↑ These are not just logically equivalent to density, they could be definitions for density, and may well be in some books.
2. ↑ [ilmath]a\in A[/ilmath] is an interior point of [ilmath]A[/ilmath] if:
   • [ilmath]\exists U\in\mathcal{J}[a\in U\wedge U\subseteq A][/ilmath] (by Functional Analysis - V1 - Dzung M. Ha - can't use references in reference tag!)
3. ↑ This is a trivial restatement of the claim and not worth going on its own page, unless it can be generalised (eg detached from dense-ness)

References

1. ↑ ^{1.0 1.1 1.2} Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
2. ↑ Introduction to Topological Manifolds - John M. Lee
3. ↑ Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp