Looking at the definition of Second-countable we can see that this weakens that notion by saying each point $x\in X$ has a countable Topological Space Basis i.e. the countable collection of neighbourhoods $\{ \mathcal{U}_{j} \}_{j\in \mathbb{N}}$ forms a neighbourhood basis in the sense that for any other neighbourhood $\mathcal{U}$ of $x$ we can find an index that points us to something in the basis s.t. $\mathcal{U}_{j}\subset \mathcal{U}$.