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Reflexive objects in topological categories

Published online by Cambridge University Press:  04 March 2009

Michael D. Rice
Michael D. Rice
Affiliation:
Computer Science Group, Mathematics Department, Wesleyan University, Middletown, CT 06459
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Abstract

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 6 , Issue 4 , August 1996 , pp. 375 - 386
Copyright
Copyright © Cambridge University Press 1996

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