Subbasis

Definition

Let $X$ be a set and let $\mathcal{S}$ be a set of subsets of $X$. The topology generated by $\mathcal{S}$ is the smallest topology on $X$ that contains $\mathcal{S}$. It is the intersection of all topologies on $X$ that contain $\mathcal{S}$. If $(X, \tau)$ is a topological space and $\tau$ is the topology generated by $\mathcal{S}$, then $\mathcal{S}$ is a subbasis of $\tau$.

Remarks

The open sets of the generated topology are the unions of finite intersections of members of $\mathcal{S}$. Here the empty intersection is $X$. Thus the finite intersections of members of $\mathcal{S}$ form a basis of the generated topology. The open rays $(-\infty, a)$ and $(b, \infty)$, with $a, b \in \mathbb{R}$, form a subbasis of the usual topology of $\mathbb{R}$. The sets $\pi_i^{-1}(U)$, with $U$ open, form a subbasis of the product topology.

Explanations

L1 human

A subbasis is a set of open sets that you want to have. The topology is the smallest one that makes them all open. You get all open sets by taking finite intersections of them and then arbitrary unions.