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A subbasis S is a collection of subsets whose finite intersections form a basis. Subbases are more economical than bases and underlie the definition of product topologies.
You already know how a basis generates a topology: take a collection of sets satisfying the basis axioms, and declare the open sets to be all possible unions of basis elements. A subbasis takes one more step back — it is a collection of sets whose finite intersections then form a basis, whose unions in turn generate the topology. Every subset of X qualifies as part of a subbasis (as long as the subbasis covers X), so subbases impose almost no conditions at all. The price for this freedom is the two-step construction: subbasis → (finite intersections) → basis → (unions) → topology.
Why introduce another layer of abstraction? Because many natural topologies are most easily described by specifying a small, convenient collection of sets that you want to be open, without worrying about closure under finite intersections upfront. The product topology is the canonical example. Given spaces X and Y, the product topology on X × Y is generated by the subbasis consisting of sets of the form U × Y (where U is open in X) and X × V (where V is open in Y). These "cylinder sets" are the natural open sets for a product: they specify a condition on one coordinate while leaving the other free. Their finite intersections are sets of the form U × V, which form the standard basis for the product topology. The subbasis makes the definition clean and conceptually transparent — you are simply saying "open sets in the product should reflect the open sets in each factor."
The subbasis perspective also clarifies what it means to specify a topology by declaring certain maps continuous. If you want a topology on a set X such that a family of maps fα: X → Yα is all continuous, the coarsest such topology has subbasis {fα⁻¹(Vα) : Vα open in Yα}. This is the initial topology (also called the weak topology in functional analysis), and it is the right topology for making the fα continuous while keeping as few open sets as possible. The subbasis is the natural language for initial topologies because you simply pull back open sets from each target space, without worrying about intersections or unions until after the subbasis is declared. In this sense, the subbasis is not just a convenient notational shortcut — it is the correct primitive for constructing topologies from families of maps.