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Published online by Cambridge University Press: 07 August 2025
We shall prove that if X, Y are compact metrizable spaces of positive dimension and ${h:X\times Y \to X}$ is a continuous map with zero-dimensional fibers then X contains a nontrivial continuum without one-dimensional subsets; in particular, X is not a countable union of zero-dimensional sets, which provides a negative answer to a question of Dudák and Vejnar [DV]