Denseness

Definition: Dense Subset

Let $(X,\tau )$ is a topological space.

A subset $S\subseteq X$ is dense iff its Closure is $X$.

$$\overline{S}=X$$

THEOREM

Let $(X,\tau )$ is a topological space.

A subset $S\subseteq X$ is dense if and only if the intersection of $S$ with each open subset is nonempty.

PROOF

TODO