Property and proper shape theory

Author:
R. B. Sher

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

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Abstract: A class of spaces called the 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 spaces in cellularity theory. Herein it is shown that the locally compact metric space *X* is an 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 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 , called , is defined and it is shown that if *X* is a closed connected subset of a piecewise linear *n*-manifold, then *X* is an space if and only if *X* is an space. Finally, it is shown that a locally finite connected simplicial complex is an space if and only if all of its homotopy and proper homotopy groups vanish.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0341389-0

Keywords:
Property ,
property ,
cellularity,
quasi-cellularity,
proper shape,
proper homotopy type,
tree,
ends,
proper homotopical domination,
property ,
property ,
proper homotopy group

Article copyright:
© Copyright 1974
American Mathematical Society