Extending homeomorphisms on Cantor cubes (Q6142393)

In this paper, the authors provide a non-metrizable analogue of the Ryll-Nardzewski theorem on extension of homeomorphisms on closed subsets of the Cantor set \(D^{\aleph_0}\) (see [\textit{B. Knaster} and \textit{M. Reichbach}, Fundam. Math. 40, 180--193 (1953; Zbl 0051.40102)]). The concept of \(\lambda\)-interior plays a key role in the proof of the main result of the paper. For a space \(X\), a subset \(P\subset X\) and an infinite cardinal \(\lambda\) the \(\lambda\)-\textit{interior} of \(P\) in \(X\), denoted by \(P^{(\lambda)}\), is the set of all \(x\in P\) such that there exists a \(G_{\lambda}\)-subset \(K\) of \(X\) with \(x\in K\subset P\). The set of all cardinals \(\lambda\) with \(P^{(\lambda)}\neq\emptyset\) is denoted by \(\mathfrak{L}_{P}\). Let \(\lambda\in\mathfrak{L}_{P}\) be a limit cardinal. The set \(\mathfrak{L}_{P}\) is said to be \textit{discontinuous at} \(\lambda\) if \(\overline{\bigcup_{\gamma<\lambda}P^{(\gamma)}}\subsetneqq P^{(\lambda)}\), otherwise \(\mathfrak{L}_{P}\) is said to be \textit{continuous at} \(\lambda\). The set of cardinals of discontinuity for \(\mathfrak{L}_{P}\) is denoted by \(\mathfrak{L}^{d}_{P}\). The set \(\mathfrak{L}^{d}_{P}\) is said to be \textit{discrete in} \(\mathfrak{L}_{P}\) if for every \(\lambda\in\mathfrak{L}^{d}_{P}\) there is \(\gamma<\lambda\) such that the interval \((\gamma,\lambda)\) is disjoint from \(\mathfrak{L}^{d}_{P}\). The main result of the paper is the following theorem. \textit{Theorem 1.} Let \(f\) be a homeomorphism between two proper closed subsets \(P\) and \(K\) of the Cantor cube \(D^{\tau}\) such that \(\mathfrak{L}^{d}_{P}\) is discrete in \(\mathfrak{L}_{P}\) and \(f(P^{(\lambda)})=K^{(\lambda)}\) for every \(\lambda\in\mathfrak{L}_P\). Then \(f\) can be extended to a homeomorphism of \(D^{\tau}\). In order to prove the main theorem, the authors first prove the following theorem and then apply it to the Cantor set. \textit{Theorem 2.} Let \(X=\prod_{\alpha\in A}X_{\alpha}\) be a product of compact metric spaces. Then the set \(\mathcal{H}(X)\) of all homeomorphisms of \(X\) equipped with the compact-open topology is an absolute extensor for zero-dimensional compact spaces. Moreover, the authors pose the next general question. \textit{Question.} Let \(f\) be a homeomorphism between two proper closed subsets \(P\) and \(K\) of \(D^{\tau}\). Is it true that \(f\) can be extended to a homeomorphism of \(D^{\tau}\) provided \(f(P^{(\lambda)})=K^{(\lambda)}\) for every \(\lambda\in\mathfrak{L}_P\)?