Compact spaces homeomorphic to their respective squares (Q6564868)

This paper deals with metrizable compacta that are homeomorphic to their squares. As pointed out by the authors, such spaces must be either \(0\)-dimensional or infinite-dimensional. They also note that in 2020 at the Wrocław Set Theory Seminar, W. Charatonik presented a joint result with Ş. Şahan in which they proved the existence of an uncountable pairwise non-homeomorphic set of \(0\)-dimensional metrizable compacta each of which is homeomorphic to its square. In the presentation, Charatonik asked if such a set having the cardinality of the continuum could be produced. The main result herein gives an affirmative answer to this question:\N\N\textbf{Theorem 1.1.} There exists a family \(\mathcal{F}\) whose cardinality is that of the continuum, whose elements are pairwise non-homeomorphic \(0\)-dimensional metrizable compacta, and which has the property that for each \(X\in\mathcal{F}\), \(X^2\) is homeomorphic to \(X\).