Dense
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 listContents
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:
- Topological: [ilmath]\forall U\in\mathcal{J}[U\cap A\neq\emptyset][/ilmath]^{[1]}
- There are some equivalent conditions^{[Note 1]}
- [ilmath]\text{Closure}(A)[/ilmath][ilmath]\eq X[/ilmath] (sometimes written: [ilmath]\overline{A}\eq X[/ilmath])
- [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]
- [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]
- 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]
- There are some equivalent conditions^{[Note 1]}
- 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.
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]}:
- [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:
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 message provided is:
This proof has been marked as an page requiring an easy proof
Claim 2
The message provided is:
This proof has been marked as an page requiring an easy proof
See also
Notes
- ↑ These are not just logically equivalent to density, they could be definitions for density, and may well be in some books.
- ↑ [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!)
- ↑ 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)