Equivalent statements to a set being dense

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


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

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

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][1][3]
    • 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 [3]

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

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[1]

Metric spaces claims


  1. In the interests of making the reader aware of caveats of formal logic as well as differences, this is equivalent to:
    1. [ilmath]\forall U\in\mathcal{J}[U\ne\emptyset\implies\exists y\in E[y\in U]][/ilmath]
    2. [ilmath]\forall U\in\mathcal{J}\exists y\in E[U\ne\emptyset\implies y\in U][/ilmath]
    3. (Obvious permutations of these)

    TODO: Show them and be certain myself. I can believe these are equivalent, but I have not shown it!


  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
  2. 2.0 2.1 Introduction to Topological Manifolds - John M. Lee
  3. 3.0 3.1 Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp