Equivalent statements to a set being dense

Stub grade: A*

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Demote to grade C or D once more theorems have been sought out and the page resembles a page rather than its current state

See Motivation:Dense set for the motivation of dense set. This page describes equivalent conditions to a set being dense.

Statements

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]E\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. Then "[ilmath]E[/ilmath] is dense in [ilmath](X,\mathcal{ J })[/ilmath]" is equivalent to any of the following:

1. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies U\cap E\ne\emptyset][/ilmath]
   • A set is dense if and only if every non-empty open subset contains a point of it - definition in
2. The closure of [ilmath]E[/ilmath] is [ilmath]X[/ilmath] itself
   • This is the definition we use and the definition given by.
3. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies \neg(U\subseteq X-E)][/ilmath] (I had to use negation/[ilmath]\neg[/ilmath] as \not{\subseteq} doesn't render well ([ilmath]\not{\subseteq} [/ilmath]))
   • [ilmath]X-E[/ilmath] contains no non-empty open set of [ilmath]X[/ilmath]
4. TODO: Symbolic form
   • [ilmath]X-E[/ilmath] has no interior points (i.e: [ilmath]\text{interior}(E)=E^\circ=\emptyset[/ilmath], the interior of [ilmath]E[/ilmath] is empty)

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TODO: Factor these out into their own pages and link to

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Metric space cases

Suppose [ilmath](X,d)[/ilmath] is a metric space and [ilmath](X,\mathcal{ J })[/ilmath] is the topological space induced by the metric space, then the following are equivalent to an arbitrary subset of [ilmath]X[/ilmath], [ilmath]E\in\mathcal{P}(X)[/ilmath] being dense in [ilmath](X,\mathcal{ J })[/ilmath]:

1. [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap E\ne\emptyset][/ilmath]
   • Words
   • This is obviously the same as: [ilmath]\forall x\in X\forall\epsilon>0\exists y\in E[y\in B_\epsilon(x)][/ilmath] - definition in

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TODO: Factor these out into their own pages and link to

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Proof of claims

Dense if and only if A set is dense if and only if every non-empty open subset contains a point of it is done already!

Grade: A

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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Would be good to at least post a picture of the work, routine and proofs are abundently available, see page 74 in