Comparing Cartesian closed categories of (core) compactly generated spaces - Escardó, Martín; Lawson, Jimmie; Simpson, Alex (Topology and its applications, 2004)
It is well known that, although the category of topological spaces is not Cartesian closed, it possesses many Cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v). We give a simple and uniform proof of Cartesian closedness for many categories of topological spaces, including (ii)–(v), and implicitly (i), and we also give a self-contained proof that (vi) is Cartesian closed. Our main aim, however, is to compare the categories (i)–(vi), and others like them. When restricted to Hausdorff spaces, (ii)–(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the Cartesian closed structure, finite products coincide in (i)–(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)–(v). Finally, the topologies of the spaces in the categories (i)–(iv) are analyzed in terms of Lawson duality.