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Transactions of the American Mathematical Society

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Pairs of compacta and trivial shape


Author: Sibe Mardešić
Journal: Trans. Amer. Math. Soc. 189 (1974), 329-336
MSC: Primary 54C56
DOI: https://doi.org/10.1090/S0002-9947-1974-0341387-7
MathSciNet review: 0341387
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Abstract: Let (X, Y, A), $ (X',Y',A')$ be triples of compact Hausdorff spaces. Using ANR-systems the following is proved: $ {\text{sh}}\;Y = {\text{sh}}\;Y' = 0,{\text{sh}}(X,Y) = {\text{sh}}(X',Y')$ and $ {\text{sh}}\;A = {\text{sh}}\;A'$ imply $ {\text{sh}}(X,Y,A) = {\text{sh}}(X',Y',A')$. All results concerning the shape of decomposition spaces and addition properties of FAR's, due to K. Borsuk and T.A. Chapman, follow readily from this theorem. In particular, $ {\text{sh}}\;X = {\text{sh}}\;X' = 0$ and $ {\text{sh}}\;A = {\text{sh}}\;A'$ imply $ {\text{sh}}(X,A) = {\text{sh}}(X',A')$, which in view of an example of Borsuk shows that for compact metric pairs the ANR-system approach to shapes differs from the Borsuk approach.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0341387-7
Keywords: Shape of pairs and triples of compacta, shape of quotient spaces, ANR-systems
Article copyright: © Copyright 1974 American Mathematical Society