Subbasis of a topology

A subbasis of a topology on a set $X$ is a collection $\mathcal{S}$ of subsets of $X$ such that the collection of all finite intersections of members of $\mathcal{S}$ forms a basis for a topology on $X$. The topology generated by $\mathcal{S}$ consists of all unions of such finite intersections.

Subbases are especially useful for defining topologies via simple building blocks; for instance, the product topology is most naturally described by a subbasis. They are also convenient for checking continuity : it often suffices to control preimages of subbasic sets.

Examples:

• In $\mathbb{R}$ with the usual topology, the family of rays $\{(-\infty,a):a\in\mathbb{R}\}\cup\{(a,\infty):a\in\mathbb{R}\}$ is a subbasis.
• For a product $X\times Y$, the sets $\pi_X^{-1}(U)$ and $\pi_Y^{-1}(V)$ (with $U$ open in $X$ and $V$ open in $Y$) form a subbasis, where $\pi_X,\pi_Y$ are the projection maps.