Definition:Everywhere Dense

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$ be a subset.


Definition 1

The subset $H$ is (everywhere) dense in $T$ if and only if:

$H^- = S$

where $H^-$ is the closure of $H$.


Definition 2

The subset $H$ is (everywhere) dense in $T$ if and only if the intersection of $H$ with every open subset of $T$ is non-empty:

$\forall U \in \tau: H \cap U \ne \varnothing$


Real Numbers

Let $S$ be a subset of the real numbers.


Then $S$ is (everywhere) dense in $\R$ if and only if:

$\forall x \in \R : \forall \epsilon \in \R : \epsilon > 0 : \exists s\in S : x-\epsilon < s < x + \epsilon$.

That is, if and only if in every neighborhood of every real number lies an element of $S$.


Also known as

Some authors refer to such a subset merely as a dense subset. However, this can be confused with dense-in-itself.


Also see

  • Results about topological denseness can be found here.