Subbase
topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T such that every open set in T can be written as a union of finite intersections of elements of B. We say that the subbase generates the topology T, and that T is generated by B. The definition is used for two purposes in practice:
- First, if a topology T on a set X is already known, then a given subcollection of T can be singled out and shown to be a subbase for T. For instance, the usual topology on the real numbers R has a subbase consisting of all semi-infinite open intervals either of the form (−∞,a) or (a,∞), where a is a real number. Note that for any given topological space X with topology T, there may be more than one subbase for T. In the example above, the set of all semi-infinite open intervals either of the form (−∞,a) or (a,∞), where a is a rational number, is also a subbase for T. And clearly, for any topological space, the entire topology is itself a subbase. Thus, a subbase for a given fixed topology is not unique.
- Secondly, given a set X, a topology T on X can be specified by giving a subbase for T. In fact, for any subcollection S of the power set P(X), there is a unique smallest topology T containing S, and this topology T is called the topology generated by S. (This topology is the intersection of all topologies on X containing S.) Compare this property to the behaviour of bases which generate topologies.
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