# 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 non-empty open set of $T$ is non-empty:

- $\forall U \in \tau \setminus \set \O: H \cap U \ne \O$

### Definition 3

The subset $H$ is **(everywhere) dense in $T$** if and only if every neighborhood of every point of $S$ contains at least one point of $H$.

## Other Abstract Spaces

### Metric Space

Let $M = \struct {A, d}$ be a metric space.

Let $B \subseteq A$ be a subset of $A$.

Then $B$ is **(everywhere) dense** in $M$ if and only if every point of $A$ is a limit point of a sequence of points of $B$.

### Normed Vector Space

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

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

Suppose:

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

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

### Real Numbers

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

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

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

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

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## Also known as

Some authors refer to an **everywhere dense** subset merely as a **dense set** or a **dense subset**.

However, this can be confused with dense-in-itself.

## Also see

- Results about
**everywhere dense**can be found**here**.