Comparing Cartesian closed categories of (core) compactly generated spaces

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 loca...

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Bibliographic Details
Published in:Topology and its applications Vol. 143; no. 1; pp. 105 - 145
Main Authors: Escardó, Martín, Lawson, Jimmie, Simpson, Alex
Format: Journal Article
Language:English
Published: Elsevier B.V 28.08.2004
Subjects:
Cartesian closed category
Compactly generated space
Core-compact space
Countably based space
Domain theory
Function space
Isbell topology
k-space
Quotient space
Compactly generated space
Cartesian closed category
Isbell topology
54D50
06B35
Function space
54C35
54D55
k-space
Countably based space
Quotient space
Domain theory
Core-compact space
ISSN:0166-8641, 1879-3207
Online Access:Get full text
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Summary: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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2004.02.011