Subbasis of a topology

A subbasis for a topology on a set $X$ is a collection $\mathcal S\subseteq\mathcal P(X)$ whose union is $X$ . The finite intersections of members of $\mathcal S$ , including the empty intersection $X$ , form a basis. The topology generated by $\mathcal S$ consists of all unions of these finite intersections and is the smallest topology containing $\mathcal S$ .

Remarks

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.