Difference between revisions of "Dense"

Latest revision as of 04:15, 1 January 2017

Temporary summary

Let $(X,\mathcal{ J })$ be a topological space, and $(X,d)$ be a metric space. Then for an arbitrary subset of $X$, say $A\in\mathcal{P}(X)$, we say $A$ is dense in $X$ if:

1. Topological: $\forall U\in\mathcal{J}[U\cap A\neq\emptyset]$^[1]
   • There are some equivalent conditions^{[Note 1]}
     1. $\text{Closure}(A)$$\eq X$ (sometimes written: $\overline{A}\eq X$)
     2. $X-A$ contains no (non-empty) open subsets of $X$
        • Symbolically: $\forall U\in\mathcal{J}[U\nsubseteq X-A]$ - which is easily seen to be equivalent to: $\forall U\in\mathcal{J}\exists p\in U[p\notin X-A]$
     3. $X-A$ has no interior points^{[Note 2]}
        • Symbolically we may write this as: $\forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right]$

          $\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)]$

          $\iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))]$ - by the negation of logical and

          $\iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A]$ - of course by the implies-subset relation we see $(A\subseteq B)\iff(\forall a\in A[a\in B])$, thus:

          $\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big]$

2. Metric: $\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap A\neq\emptyset$
   • There are no equivalent statements at this time.

The rest of the page continues below. It will be refactored soon.

Definition

Let $(X,\mathcal{ J })$ be a topological space and let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$. We say "$A$ is dense in $X$ if^[2]:

• $\overline{A}=X$ - that is to say that the closure of $A$ is the entirety of $X$ itself.

Some authors give the following equivalent definition to $A$ being dense^[1]:

• $\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies y\in U]$, which is obviously equivalent to: $\forall U\in\mathcal{J}[U\ne\emptyset \implies A\cap U\ne\emptyset]$ (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 $(X,d)$ me a metric space, we say that $E\in\mathcal{P}(X)$ (so $E$ is an arbitrary subset of $X$) if^[1]:

• $\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset]$ - where $B_r(x)$ denotes the open ball of radius $r$, centred at $x$
  • In words: Every open ball at every point overlaps with $E$. (i.e: every open ball at every point contains at least 1 point in common with $E$)
  • This is equivalent to $\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)]$^{Found in:[3]} (see Claim 1)^{[Note 3]}

Claim 2: for a metric space $(X,d)$ a subset, $E\in\mathcal{P}(X)$ is dense in the metric sense if and only if it is dense in $(X,\mathcal{ J })$ where $J$ is the topology induced by the metric $d$.

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. Jump up ↑ These are not just logically equivalent to density, they could be definitions for density, and may well be in some books.
2. Jump up ↑ $a\in A$ is an interior point of $A$ if:
   • $\exists U\in\mathcal{J}[a\in U\wedge U\subseteq A]$ (by Functional Analysis - V1 - Dzung M. Ha - can't use references in reference tag!)
3. Jump up ↑ 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. ↑ ^{Jump up to: 1.0 1.1 1.2} Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
2. Jump up ↑ Introduction to Topological Manifolds - John M. Lee
3. Jump up ↑ Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp