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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Property $SUV^{\infty }$ and proper shape theory
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by R. B. Sher PDF
Trans. Amer. Math. Soc. 190 (1974), 345-356 Request permission

Abstract:

A class of spaces called the $SU{V^\infty }$ spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the $U{V^\infty }$ spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an $SU{V^\infty }$ space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the $SU{V^\infty }$ spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property $U{V^n}$, called $SU{V^n}$, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an $SU{V^n}$ space if and only if X is an $SU{V^\infty }$ space. Finally, it is shown that a locally finite connected simplicial complex is an $SU{V^\infty }$ space if and only if all of its homotopy and proper homotopy groups vanish.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 190 (1974), 345-356
  • MSC: Primary 54C56
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0341389-0
  • MathSciNet review: 0341389