Continuity is an Adjoint Functor

1. INTRODUCTION. The emergence of category theory, introduced by S. Eilen-berg and S. Mac Lane in the 1940s (cf. [2]), was among the most important mathematical developments of the twentieth century. The profound impact of the theory continues to this day, and categorical methods are currently used, for example, in algebra, geometry, topology, mathematical physics, logic, and theoretical computer science. (Hints of this breadth can be found, e.g., in [1], [6], and [7]). A category is comprised of objects and morphisms betweeen objects. Standard elementary examples include Set, where the objects are sets and the morphisms are set functions; Grp, where the objects are groups and the morphisms are group homo-morphisms; and T op, where the objects are topological spaces and the morphisms are continuous functions. Relationships among different categories are established via functors between them. A pair of adjoint functors, as formulated in 1958 by D. M. Kan [3], determines a particularly close tie between two categories. Adjoint functors are essential tools in category theory, and their introduction was a significant milestone in its development. While it is commonly held that adjoint functors " occur almost everywhere " [6, p. 107]), at least in many areas of mathematics, the typical first examples presented to students may not immediately reveal the fundamental importance of the ideas involved. (One such typical example, a left adjoint to a " forgetful functor, " is described at the end of the brief review provided in the next section.) Our aim in this note, then, is to illustrate how a natural example of adjoint functors can be " found " in the definition of a continuous map between topological spaces. In particular, we show, for a set function ϕ : X → Y between topological spaces X and Y , that ϕ is continuous if and only if certain naturally arising functors are adjoint.