Subbase

In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. Let be a topological space with topology A subbase of is usually defined as a subcollection of satisfying one of the two following equivalent conditions: The subcollection generates the topology This means that is the smallest topology containing : any topology on containing must also contain The collection of open sets consisting of all finite intersections of elements of together with the set forms a basis for This means that every proper open set in can be written as a union of finite intersections of elements of Explicitly, given a point in an open set there are finitely many sets of such that the intersection of these sets contains and is contained in (If we use the nullary intersection convention, then there is no need to include in the second definition.) For subcollection of the power set there is a unique topology having as a subbase. In particular, the intersection of all topologies on containing 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 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. Less commonly, a slightly different definition of subbase is given which requires that the subbase cover In this case, is the union of all sets contained in This means that there can be no confusion regarding the use of nullary intersections in the definition. However, this definition is not always equivalent to the two definitions above.