Definition:Everywhere Dense/Definition 1

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$ be a subset.

The subset $H$ is (everywhere) dense in $T$ if and only if:

$H^- = S$

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

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

Also see

• Equivalence of Definitions of Everywhere Dense

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.20$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Countability Properties