Pairs of compacta and trivial shape

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.