In topology, a subbase for a topological space with topology is a subcollection of that generates, in the sense that is the smallest topology containing. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
Definition
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 setsconsisting 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.
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.
Alternative definition
Sometimes, 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, with this definition, the two definitions above are not always equivalent. In other words, there exist spaces with topology, such that there exists a subcollection of such that is the smallest topology containing, yet does not cover. In practice, this is a rare occurrence; e.g. a subbase of a space that has at least two points and satisfies the T1separation axiom must be a cover of that space.
Examples
The usual topology on the real numbers has a subbase consisting of all semi-infinite open intervals either of the form or, where and are real numbers. Together, these generate the usual topology, since the intersections for generate the usual topology. A second subbase is formed by taking the subfamily where and are rational. The second subbase generates the usual topology as well, since the open intervals with, rational, are a basis for the usual Euclidean topology. The subbase consisting of all semi-infinite open intervals of the form alone, where 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. The initial topology on defined by a family of functions, where each has a topology, is the coarsest topology on such that each is continuous. Because continuity can be defined in terms of the inverse images of open sets, this means that the initial topology on is given by taking all, where ranges over all open subsets of, as a subbasis. Two important special cases of the initial topology are the product topology, where the family of functions is the set of projections from the product to each factor, and the subspace topology, where the family consists of just one function, the inclusion map. The compact-open topology on the space of continuous functions from to has for a subbase the set of functions where is compact and is an open subset of.
Results using subbases
One nice fact about subbases is that continuity of a function need only be checked on a subbase of the range. That is, if is a subbase for, a function is continuous iff is open in for each in.
Alexander subbase theorem
There is one significant result concerning subbases, due to James Waddell Alexander II. Note that the corresponding result for basic covers is trivial. Although this proof makes use of Zorn's Lemma, the proof does not need the full strength of choice. Instead, it relies on the intermediate Ultrafilter principle. Using this theorem with the subbase for above, one can give a very easy proof that bounded closed intervals in are compact. Tychonoff's theorem, that the product of compact spaces is compact, also has a short proof. The product topology on has, by definition, a subbase consisting of cylinder sets that are the inverse projections of an open set in one factor. Given a subbasic family of the product that does not have a finite subcover, we can partition into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space. By assumption, no has a finite subcover. Being cylinder sets, this means their projections onto have no finite subcover, and since each is compact, we can find a point that is not covered by the projections of onto. But then is not covered by. Note, that in the last step we implicitly used the axiom of choice to ensure the existence of.
