Topological representations of bornological spaces and coercive maps

Abstract

We exhibit the duality between bornological spaces and a certain class of topological spaces of independent interest: the topological spaces of the cofinite type. The usual way to make the bornological spaces a category Bor is to take for morphisms bornological maps [9]. If we instead take coercive maps as discussed in [5] as morphisms, we obtain a very different category Bor^⁎ that is isomorphic to the category of topological spaces of the cofinite type equipped with continuous maps as morphisms. We introduce a bornology on the bornological maps bor$(X,Y)$ between bornological spaces $(X,B)$ and $(Y,C)$. Membership of $E\subseteq \text{bor}(X,Y)$ to this bornology means that the bornological maps in E are subject to a uniform growth bound. This growth bound can be expressed in terms of an expansion modulus, i.e., an increasing function between $B$ and $C$ mapping nonempty members of $B$ to nonempty members of $C$. We introduce a parallel bornology on the coercive maps cocv$(X,Y)$ between the bornological spaces as well. Notably, evaluation as defined on bor$(X,Y)\times X$ is a bornological map, whereas evaluation may fail to be a coercive map on cocv$(X,Y)\times X$.