Definition:Everywhere Dense

Definition

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

Let $H \subseteq S$.

Then $H$ is (everywhere) dense in $T$ iff:

$H^- = S$

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

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

Also see

• Nowhere dense
• Dense-in-itself

• Results about topological denseness can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Countability Properties