Vol. 88, No. 1, 1980

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Monoidal closed, Cartesian closed and convenient categories of topological spaces

Peter I. Booth and J. Tillotson

Vol. 88 (1980), No. 1, 35–53

This paper develops a unified theory of function spaces M𝒜(Y,Z) with set-open topologies, the sets in question being the continuous images of selected classes of topological spaces 𝒜. We prove that at least five of these function spaces are distinct and have corresponding exponential homeomorphisms 𝜃 : M𝒜(X,M𝒜(Y,Z))M𝒜(X ×𝒜Y,Z) for suitably retopologized product spaces X ×𝒜Y . Singleton spaces are normally identities with respect to these products and so we have determined four distinct monoidal closed structures for the category of all spaces. Conditions for the category of spaces generated by 𝒜, i.e., the coreflective hull of 𝒜, to be cartesian closed and/or convenient are given. One result asserts that the category of sequential spaces is the smallest convenient category.

Mathematical Subject Classification 2000
Primary: 55U40
Secondary: 18B30, 18D15, 54B10, 54C35
Received: 2 October 1978
Revised: 29 June 1979
Published: 1 May 1980
Peter I. Booth
J. Tillotson