Definition (Dense)
A subset A⊂X of a topological space is dense if cl(A)=X.
Definition (Dense (metric space))
Let B be a Metric Space, and let A⊂B. A is dense in B if: ∀b∈B,ϵ>0,∃a∈A:∥a−b∥≤ϵ
Intuition
An analogous way of stating this is saying that there exist sequences (an)n∈N⊂A s.t. limn→∞an∈B.