Continuity is an adjoint functor

TL;DR

This paper demonstrates that a function between topological spaces is continuous if and only if it induces a pair of adjoint functors between categories of closed sets, providing elementary examples of such adjoint pairs.

Contribution

It establishes a novel characterization of continuity via adjoint functors between categories of closed subsets, enriching the categorical understanding of topological continuity.

Findings

Continuity corresponds to a pair of adjoint functors between closed set categories.

Provides elementary examples of adjoint pairs not typically covered in standard courses.

Connects topological continuity with categorical adjunctions in a novel way.

Abstract

For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X \rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (apparently) not part of the standard introductory treatments of this subject.