Difference between revisions of "Dense"
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| − | {{Stub page|grade=B|msg=Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote}} | + | {{Stub page|grade=B|msg=Revise page, add some links to propositions or theorems using the dense property. Also more references, then demote.<br/> |
| + | '''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== | ==Definition== | ||
Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. We say "''{{M|A}} is dense in {{M|X}}'' if{{rITTMJML}}: | Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset]] of {{M|X}}. We say "''{{M|A}} is dense in {{M|X}}'' if{{rITTMJML}}: | ||
Revision as of 10:27, 30 December 2016
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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 listContents
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[1]:
- [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[2]:
- [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[2]:
- [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 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:
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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.
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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
Notes
- ↑ 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)
