Compact spaces homeomorphic to their respective squares
Jan Dudák, Benjamin Vejnar
TL;DR
The paper proves that there exists a continuum-sized family of pairwise non-homeomorphic compact metrizable zero-dimensional spaces $X$ with $X\times X\cong X$, addressing a question of Charatonik. It develops multiple construction strategies for infinite-dimensional continua with the same square property, including graph-based universality, Cook continua, and algebraic-topology methods, and extends some ideas to Peano continua and absolute retracts. A central contribution is the explicit zero-dimensional construction via $X(M)$ spaces and the $\mathscr{S}(M)$ framework, yielding a rich, non-redundant catalog of spaces with the square property. The final section corrects a key step in CharatonikSahan, supplying a robust ordinal-indexed partition and extension framework that ensures correct extension of homeomorphisms from perfect kernels to the entire space. Together, the results illuminate the landscape of spaces that are homeomorphic to their squares across dimensions and categories, with implications for topology and set-theoretic topology.
Abstract
We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of W. J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.