Difference between revisions of "Dense"

Revision as of 19:40, 28 October 2016

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^[1]:

• $\overline{A}=X$ - 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^[2]:

• $\forall U\in\mathcal{J}\exists a\in A[U\ne\emptyset\implies y\in U]$, 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^[2]:

• $\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset]$ - 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 1]}

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 ↑ 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 ↑ Introduction to Topological Manifolds - John M. Lee
2. ↑ ^{Jump up to: 2.0 2.1} Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
3. Jump up ↑ Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp