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

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

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

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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