Cartesian closed coreflective subcategories of topological spaces determined by monotone convergence classes

TL;DR

This paper characterizes certain subcategories of topological spaces that are cartesian closed, using monotone convergence classes and the concept of $ ext{P}$-determined spaces, with applications to domain theory.

Contribution

It introduces $ ext{P}$-determined spaces and shows that categories of such spaces are exactly the coreflective subcategories of $f Top$, proving their cartesian closedness.

Findings

Categories of $ ext{P}$-determined spaces are cartesian closed.

Exponential objects in these categories relate to those in domain theory.

Several classes of spaces determined by monotone convergence are analyzed in detail.

Abstract

We introduce the notion of an operation $\mathcal{P}$ and a $\mathcal{P}$-determined space. It is shown that a category is a coreflective full subcategory of $\bf Top$ if and only if it is equal to $\bf Top_{\mathcal{P}}$ for some idempotent and consistent operation $\mathcal{P}$, where $\bf Top_{\mathcal{P}}$ is the category of all $\mathcal{P}$-determined spaces. As concrete examples of $\mathcal{P}$-determined spaces, several classes of topological spaces determined by monotone convergence classes are investigated in detail. By the tool of $\mathbb{C}$-generated spaces, it is shown uniformly that categories of these examples of $\mathcal{P}$-determined spaces are all cartesian closed. Moreover, the exponential objects and categorical products of some categories in domain theory are shown to be closely related to those of $\bf DTop$, the category of directed spaces.