Topological and limit-space subcategories of countably-based equilogical spaces

Matias Menni, Alex Simpson

Research output: Contribution to journalArticlepeer-review

Abstract

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. In fact, this category turns out to be equivalent to the category of all quotient spaces of countably based topological spaces. We show that the category is bicartesian 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.
Original languageEnglish
Pages (from-to)739-770
Number of pages32
JournalMathematical Structures in Computer Science
Volume12
DOIs
Publication statusPublished - 1 Dec 2002

Fingerprint

Dive into the research topics of 'Topological and limit-space subcategories of countably-based equilogical spaces'. Together they form a unique fingerprint.

Cite this