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General function spaces, products and continuous lattices

Published online by Cambridge University Press:  24 October 2008

John Isbell
John Isbell
Affiliation:
State University of New York at Buffalo, USA
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Extract

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 2 , September 1986 , pp. 193 - 205
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1] Arens, R.. A topology for spaces of transformations. Ann. of Math. (2) 47 (1946), 480–495.CrossRefGoogle Scholar
[2] Arens, R. and Dugundji, J.. Topologies for function spaces. Pacific J. Math 1 (1951), 5–31.CrossRefGoogle Scholar
[3] Booth, P. and Tillotson, J.. Monoidal closed, cartesian closed and convenient categories of topological spaces. Pacific J. Math. 88 (1980), 35–53.CrossRefGoogle Scholar
[4] Brown, R.. Function spaces and product topologies. Quart. J. Math. (Oxford) (2) 15 (1964), 238–250.CrossRefGoogle Scholar
[5] Činčura, J.. Closed structures on categories of topological spaces. (To appear.)Google Scholar
[6] Cook, C. and Fischer, H.. On equicontinuity and continuous convergence. Math. Ann. 159 (1965), 94–104.CrossRefGoogle Scholar
[7] Day, A.. Filter monads, continuous lattices and closure systems. Canadian J. Math. 27 (1975), 50–59.CrossRefGoogle Scholar
[8] Day, B. J.. Relationship of Spanier's quasi-topological spaces to k-spaces. M.Sc. thesis, University of Sydney, 1968.Google Scholar
[9] Day, B. J.. A reflection theorem for closed categories. J. Pure Appl. Algebra 2 (1972), 1–11.CrossRefGoogle Scholar
[10] Day, B. J. and Kelly, G. M.. On topological quotient maps preserved by pullbacks or products. Proc. Cambridge Philos. Soc. 67 (1970), 553–558.CrossRefGoogle Scholar
[11] Eilenberg, S. and Kelly, G. M.. Closed categories. Proc. Conference on Categorical Algebra, La Jolla 1965 (Springer-Verlag, 1966), 421–562.Google Scholar
[12] Eilenberg, S. and Steenrod, N.. Foundations of Algebraic Topology. (Princeton University Press, 1952).CrossRefGoogle Scholar
[13] Erné, M. and Gatzke, H.. Convergence and continuity in partially ordered sets and semilattices. Continuous Lattices and Their Applications (Dekker, 1985), 9–40.Google Scholar
[14] Fell, J. M. G.. A Hausdorff topology for the closed sets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc. 13 (1962), 472–476.CrossRefGoogle Scholar
[15] Fischer, H.. Limesräume. Math. Ann. 137 (1959), 269–303.CrossRefGoogle Scholar
[16] Fox, R. H.. On topologies for function spaces. Bull. Amer. Math Soc. 51 (1945), 429–432.CrossRefGoogle Scholar
[17] Freyd, P.. Abelian Categories (Harper and Row, 1964).Google Scholar
[18] Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S.. A Compendium of Continuous Lattices (Springer-Verlag, 1980).CrossRefGoogle Scholar
[19] Gierz, G. and Lawson, J. D.. Generalized continuous and hypercontinuous lattices. Rocky Mountain J. Math. 11 (1981), 271–296.CrossRefGoogle Scholar
[20] Greve, G.. How many monoidal closed structures are there in Top? Arch. Math. 34 (1980), 538–539.CrossRefGoogle Scholar
[21] Hofmann, K. H. and Lawson, J. D.. The spectral theory of distributive continuous lattices. Trans. Amer. Math Soc. 246 (1978), 285–310.CrossRefGoogle Scholar
[22] Hofmann, K. H. and Stralka, A. R.. The algebraic theory of compact Lawson semilattices. Diss. Math 137 (1976), 1–54.Google Scholar
[23] Hyland, J. M. E.. Function spaces in the category of locales. Proc. Conference on Topological and Categorical Aspects of Continuous Lattices, Bremen 1979, Springer Lecture Notes 871, 1981, 264–281.CrossRefGoogle Scholar
[24] Isbell, J.. Atomless parts of spaces. Math. Scand. 31 (1972), 5–32.CrossRefGoogle Scholar
[25] Isbell, J.. Meet-continuous lattices. Symposia Math. 16 (1975), 41–54.Google Scholar
[26] Isbell, J.. Function spaces and adjoints. Math. Scand. 36 (1975), 317–339.CrossRefGoogle Scholar
[27] Isbell, J.. Discontinuity of meets and joins. Continuous Lattices and Their Applications (Dekker, 1985), 151–154.Google Scholar
[28] James, I. M.. General Topology and Homotopy Theory. (Springer-Verlag, 1984).CrossRefGoogle Scholar
[29] Johnstone, P. T.. Stone Spaces (Cambridge University Press, 1982).Google Scholar
[30] Johnstone, P. T.. Open locales and exponentiation. Mathematical Applications of Category Theory, Contemporary Mathematics vol. 30 (American Mathematical Society, 1984), 84–116.CrossRefGoogle Scholar
[31] Johnstone, P. T.. Vietoris locales and localic somilattices. Continuous Lattices and Their Applications (Dekker, 1985), 155–180.Google Scholar
[32] Johnstone, P. T. and Joyal, A.. Continuous categories and exponentiable toposes. J. Pure Appl. Algebra 25 (1982), 255–296.CrossRefGoogle Scholar
[33] Lambrinos, P. Th. and Papadopoulos, B.. The (strong) Isbell topology and (weakly) continuous lattices. Continuous Lattices and Their Applications (Dekker, 1985), 191–211.Google Scholar
[34] Lawson, J. D.. Topological semilattices with small semilattices. J. London Math. Soc. 1 (1969), 719–724.CrossRefGoogle Scholar
[35] Lawson, J. D.. Lattices with no interval homomorphisms. Pacific J. Math. 32 (1970), 459–465.CrossRefGoogle Scholar
[36] Lawson, J. D.. Duality of continuous posets. Houston J. Math. 5 (1979), 357–394.Google Scholar
[37] Lea, J. W. Jr. Continuous lattices and compact Lawson semilattices. Semigroup Forum 13 (1976), 387–388.CrossRefGoogle Scholar
[38] Lewis, L. G. Jr. Open maps, colimits, and a convenient category of fibre spaces. Topology Appl. 19 (1985), 75–89.CrossRefGoogle Scholar
[39] Markowsky, G.. A motivation and generalization of Scott's notion of a continuous lattice. Proc. Conference on Togological and Categorical Aspecte of Continuous Lattices, Bremen 1979. Springer Lecture Notes 871, 1981, 298–307.CrossRefGoogle Scholar
[40] Michael, E.. Bi-quotient maps and Cartesian products of quotient maps. Ann. Inst. Fourier (Grenoble) 18 (1968), 287–302.CrossRefGoogle Scholar
[41] Nel, L. D.. Cartesian closed corefiective hulls. Quaestiones Math. 2 (1977), 269–283.CrossRefGoogle Scholar
[42] Niefield, S. B.. Cartesianness: topological spaces, uniform spaces and affine schemes. J. Pure Appl. Algebra 23 (1982), 147–167.CrossRefGoogle Scholar
[43] Niefield, S. B.. Cartesian spaces over T and locales over ω(T). Cahiers Topologie Géom. Differentielle 23 (1982), 257–267.Google Scholar
[44] Niefield, S. B.. Cartesian inclusions: locales and toposes. Comm. Algebra 9 (16) (1981), 1639–1671.CrossRefGoogle Scholar
[45] Pedicchio, M. C.. On the category of topological topologies. Cahiers Topologie Géom. Differentielle 25 (1984), 3–13.Google Scholar
[46] Penicchio, M. C. and Solimini, S.. On a ‘good’ dense class of topological spaces. (Preprint, Univ. di Trieste, 1984).Google Scholar
[47] Schwarz, F.. Powers and exponential objects in initially structured categories and applications to categories of limit spaces. Quaestiones Math. 6 (1983), 227–254.CrossRefGoogle Scholar
[48] Schwarz, F.. When is continuous convergence topological? Abstracts Amer. Math. Soc. 6 (1985), 89.Google Scholar
[49] Schwarz, F. and Weck, S.. Scott topology, Isbell topology and continuous convergence. Continuous Lattices and Their Applications (Dekker, 1985), 251–272.Google Scholar
[50] Scott, D. S.. Continuous lattices. Toposes, Algebraic Geometry and Logic, Halifax 1971. Springer Lecture Notes 274, 1972, 97–136.CrossRefGoogle Scholar
[51] Spanier, E. H.. Quasi-topologies. Duke Math. J. 30 (1963), 1–14.CrossRefGoogle Scholar
[52] Steenrod, N.. A convenient category of topological spaces. Michigan Math. J. 14 (1967), 133–152.CrossRefGoogle Scholar
[53] Stoy, J. E.. Denotational Semantics – The Scott–Strachey Approach to Programming Language Theory. (MIT, 1977.)Google Scholar
[54] Vietoris, L.. Bereiche zweiter Ordnung. Monatsh. Math Phys. 32 (1922), 258–280.CrossRefGoogle Scholar