Dense

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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 list

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:

• $\overline{A}=X$ - 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:

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

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

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.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

It is obvious that $(B_\epsilon(x)\cap E\ne\emptyset)\iff(\exists y\in E[y\in B_\epsilon(x)])$

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.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

Easy for someone informed of what a metric space and topology is. The claim means showing that if $(X,\mathcal{ J })$ is the topological space induced by a metric space for a metric space $(X,d)$ then $E$ is dense in $(X,\mathcal{ J })$ if and only if $E$ is dense in $(X,d)$

This proof has been marked as an page requiring an easy proof

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