On the Construction of New Topological Spaces from Existing Ones

In this note, we introduce a guiding principle to define topologies for a wide variety of spaces built from existing topological spaces. The topologies so-constructed will have a universal property taking one of two forms. If the topology is the coarsest so that a certain condition holds, we will give an elementary characterization of all continuous functions taking values in this new space. Alternatively, if the topology is the finest so that a certain condition holds, we will characterize all continuous functions whose domain is the new space. Consider a function f : X → Y between a pair of sets. If Y is a topological space, we could define a topology on X by asking that it is the coarsest topology so that f is continuous. (The finest topology making f continuous is the discrete topology.) Explicitly, a subbasis of open sets of X is given by the preimages of open sets of Y . With this definition, a function W → X, where W is some other space, is continuous if and only if the composite function W → Y is continuous. On the other hand, if X is assumed to be a topological space, we could define a topology on Y by asking that it is the finest topology so that f is continuous. (The coarsest topology making f continuous is the indiscrete topology.) Explicitly, a subset of Y is open if and only if its preimage in X is open. With this definition, a function Y → Z, where Z is some other space, is continuous if and only if the composite function X → Z is continuous. In this what follows, we use a mild generalization of this principle (where single maps are replaced by families of maps) to define topologies on new spaces constructed from old. We first describe the construction of the underlying sets of these spaces. 1. On the construction of new sets There are many ways to build new sets from existing sets: Disjoint unions. Given sets A and B, their disjoint union is the set A ∐ B whose elements are elements of exactly one of A or B. For example, the integers Z are the disjoint union of the odd integers and the even integers. Or, iterating the disjoint union construction, the set of n elements is the disjoint union of n copies of the set ∗ with a single element. Products. Given sets A and B their (cartesian) product is the set A×B = {(a, b) | a ∈ A, b ∈ B} Date: March 23, 2015. 1