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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Cartesian-closed coreflective subcategories of uniform spaces


Authors: M. D. Rice and G. J. Tashjian
Journal: Trans. Amer. Math. Soc. 276 (1983), 289-300
MSC: Primary 54E15; Secondary 18B30, 18D15, 54B30
DOI: https://doi.org/10.1090/S0002-9947-1983-0684509-6
MathSciNet review: 684509
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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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DOI: https://doi.org/10.1090/S0002-9947-1983-0684509-6
Article copyright: © Copyright 1983 American Mathematical Society