Category:Definitions/Everywhere Dense

This category contains definitions related to Everywhere Dense.
Related results can be found in Category:Everywhere Dense.

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

Let $H \subseteq S$ be a subset.

Definition 1

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

$H^- = S$

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

Definition 2

The subset $H$ is (everywhere) dense in $T$ if and only if the intersection of $H$ with every non-empty open set of $T$ is non-empty:

$\forall U \in \tau \setminus \set \O: H \cap U \ne \O$

Definition 3

The subset $H$ is (everywhere) dense in $T$ if and only if every neighborhood of every point of $S$ contains at least one point of $H$.