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Frölicher versus Differential Spaces: A Prelude to Cosmology

In: Papers in Honour of Bernhard Banaschewski

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  • Paul Cherenack

    (University of Cape Town, Department of Mathematics and Applied Mathematics)

Abstract

Differential spaces due to R. Sikorski are compared with Frölicher spaces (earlier called smooth spaces and studied by A. Frölicher). The category of Frölicher spaces is known to be topological over sets and Cartesian closed under the usual set adjunction. We see that the category of differential spaces is topological over sets but not Cartesian closed under the usual set adjunction. The tangent and cotangent bundles of an arbitrary Frölicher space X are defined and these notions are seen to coincide with the usual ones if X is a smooth manifold. We show that one can define, in a fairly natural way, the notions metric, connection, Ricci tensor, Riemannian curvature tensor and finally Einstein tensor for an arbitrary Frölicher space.

Suggested Citation

  • Paul Cherenack, 2000. "Frölicher versus Differential Spaces: A Prelude to Cosmology," Springer Books, in: Guillaume Brümmer & Christopher Gilmour (ed.), Papers in Honour of Bernhard Banaschewski, pages 391-413, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-2529-3_24
    DOI: 10.1007/978-94-017-2529-3_24
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