Definition:Nowhere Dense/Definition 1

Definition

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

Let $H \subseteq S$.

$H$ is nowhere dense in $T$ if and only if:

$\paren {H^-}^\circ = \O$

where $H^-$ denotes the closure of $H$ and $H^\circ$ denotes its interior.

That is, $H$ is nowhere dense in $T$ if and only if the interior of its closure is empty.

Another way of putting it is that $H$ is nowhere dense in $T$ if and only if it consists entirely of boundary.

Also see

• Equivalence of Definitions of Nowhere Dense

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.26$
• 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
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $22.1$: The Baire Category Theorem