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.
Categories: math.GN math.CT