Subbase

In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. Let X be a topological space with topology T. A subbase of T is usually defined as a subcollection B of T satisfying one of the two following equivalent conditions: (Note that if we use the nullary intersection convention, then there is no need to include X in the second definition.) For any subcollection S of the power set P(X), there is a unique topology having S as a subbase. In particular, the intersection of all topologies on X containing S satisfies this condition. In general, however, there is no unique subbasis for a given topology. Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set P(X) and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other. Sometimes, a slightly different definition of subbase is given which requires that the subbase B cover X. In this case, X is the union of all sets contained in B. This means that there can be no confusion regarding the use of nullary intersections in the definition. However, with this definition, the two definitions above are not always equivalent. In other words, there exist spaces X with topology T, such that there exists a subcollection B of T such that T is the smallest topology containing B, yet B does not cover X. In practice, this is a rare occurrence; e.g. a subbase of a space that has at least two points and satisfies the T1 separation axiom must be a cover of that space. The usual topology on the real numbers R has a subbase consisting of all semi-infinite open intervals either of the form (−∞,a) or (b,∞), where a and b are real numbers. Together, these generate the usual topology, since the intersections (a,b) = (−∞,b) ∩ (a,∞) for a < b generate the usual topology. A second subbase is formed by taking the subfamily where a and b are rational. The second subbase generates the usual topology as well, since the open intervals (a,b) with a, b rational, are a basis for the usual Euclidean topology. The subbase consisting of all semi-infinite open intervals of the form (−∞,a) alone, where a is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom, since all open sets have a non-empty intersection.