Definition:Nowhere Dense

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Definition

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

Let $H \subseteq S$.


Definition 1

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

$\left({H^-}\right)^\circ = \varnothing$

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


  • Results about topological denseness can be found here.


Examples