Dense Set

Definition

a subset $A$ of a Topological Space $(X,\mathcal{T})$ is dense in $X$ if the Closure of $A$ in $X$ is equal to $X.$ $\bar{A}=X$

Or, equivalently, for every element of $X$, every Neighborhood of $x$ intersects $A$. $\forall x\in X,\forall U:={\mathcal{N}}_x,U\cap A\ne \varnothing$

Denseness of the Rational Numbers

$\forall x,y\in \mathbb{R},a<b\Rightarrow \exists r\in \mathbb{Q},a<r<b$

Denseness of the Irrational Numbers

$\forall x,y\in \mathbb{R},a<b\Rightarrow \exists s\in \mathbb{I},a<s<b$