Definition:Everywhere Dense

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 $T$ :
 * $H^- = S$

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

That is, every point in $S$ is a point or a limit point of $H$.

Definition 2
The subset $H$ is (everywhere) dense in $T$ $H$ has nonempty intersection with every open subset of $T$:
 * $\forall U \in \tau : H\cap U \neq \varnothing$

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

 * Definition:Nowhere Dense
 * Definition:Dense-in-itself