Definition:Nowhere Dense

Definition

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

Let $H \subseteq S$.

Definition 1

$H$ 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.

Definition 2

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

$H^-$ contains no open set of $T$ which is non-empty

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

Also see

• Equivalence of Definitions of Nowhere Dense

• Definition:Everywhere Dense
• Definition:Dense-in-itself

• Results about topological denseness can be found here.

Examples

• Set of Reciprocals of Positive Integers is Nowhere Dense in Reals