Cartesian-closed coreflective subcategories of uniform spaces

Rice, M. D.; Tashjian, G. J.

doi:10.1090/S0002-9947-1983-0684509-6

Cartesian-closed coreflective subcategories of uniform spaces
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by M. D. Rice and G. J. Tashjian PDF

Trans. Amer. Math. Soc. 276 (1983), 289-300 Request permission

Abstract:

This paper characterizes the coreflective subcategories $\mathcal {C}$ of uniform spaces for which a natural function space structure generates the exponential law ${X^{Y \otimes Z}} = {({X^Y})^Z}$ on $\mathcal {C}$. Such categories are cartesian-closed. Specifically, we show that $\mathcal {C}$ is cartesian-closed in this way if and only if $\mathcal {C}$ is inductively generated by a finitely productive family of locally fine spaces. The results divide naturally into two cases: those subcategories containing the unit interval are generated by precompact spaces, while the subcategories not containing the unit interval are generated by spaces which admit an infinite cardinal. These results may be used to derive the characterizations of cartesian-closed coreflective subcategories of Tychonoff spaces found in [10].

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