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.