Definition:Everywhere Dense

Definition

Let $T = \struct {S, \tau}$ 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:

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

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 $S$.

Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Let $Y \subseteq X$ be a subset of $X$.

Suppose:

$\forall x \in X : \forall \epsilon \in \R : \epsilon > 0 : \exists y \in Y : \norm {x - y} < \epsilon$

Then $Y$ is (everywhere) dense in $X$.

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.