Topological and Limit-Space Subcategories of Countably-Based Equilogical Spaces

There are two main approaches to obtaining “topological” cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed — for example, the category of sequential spaces. Under the other, one generalises the notion of space — for example, to Scott’s notion of equilogical space. In this paper we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably-based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. This category consists of certain “ω-projecting” topological quotients of countably-based topological spaces, and contains, in particular, all countably-based spaces. We show that this category is cartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the category of equilogical spaces. We also show that the category of countably-based equilogical spaces has a larger full subcategory that can be simultaneously viewed as a full subcategory of limit spaces. This full subcategory is locally cartesian closed and the embeddings into limit spaces and countably-based equilogical spaces preserve this structure. We observe that it seems essential to go beyond the realm of topological spaces to achieve this result.