A 2-dimensional compactum in the product of two 1-dimensional compacta which does not contain any rectangle by Roman Pol

Electronic Version 16 (1991), 133-135

A 2-dimensional compactum in the product of two 1-dimensional compacta which does not contain any rectangle

Roman Pol

Our terminology follows Kuratowski. By a compactum we mean a compact metrizable space.

The aim of this note is to provide the following example which answers (for n = 2) a question asked by Y. Sternfeld.

Example For each natural n >= 2 there exist 1-dimensional compacta Z_i, i = 1, ... , n, and an n-dimensional compactum Y in the product Z₁ × ... ×Z_n such that whenever A₁ × ... ×A_n 'subset' Y, then all but one A_i are singletons.

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