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On decomposition of polyhedra into a Cartesian product of 1-dimensional and 2-dimensional factors. (English) Zbl 0872.57029

K. Borsuk [Fundam. Math. 31, 137-148 (1938; Zbl 0019.28201)] proved that the decomposition of a polyhedron into a Cartesian product of 1-dimensional factors is topologically unique (up to a permutation of the factors). We prove a little more general
Theorem. If a connected polyhedron \(K\) (of arbitrary dimension) is homeomorphic to a Cartesian product \(A_1 \times \cdots \times A_n\), where the \(A_i\)’s are prime compacta of dimension at most 1, then there is no other topologically different system of prime compacta \(Y_1, \dots, Y_k\) of dimension at most 2 such that \(Y_1 \times \cdots \times Y_k\) is homeomorphic to \(K\).
A space \(X\) is said to be prime if it has more than one point and only \(X\) and the singleton as Cartesian factors.

MSC:

57Q05 General topology of complexes

Citations:

Zbl 0019.28201
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