Every finite-dimensional analytic space is \(\sigma \)-homogeneous (Q6592021)

The paper investigates under the assumption of ZF+DC the \(\sigma\)-homogeneity of separable metrizable topological spaces (only such spaces are considered in the text). A space is called homogeneous if for every \(x,y\in X\) there exists a homeomorphism \(h\colon X\to X\) such that \(h(x)=y\). A space is \(\sigma\)-homogeneous if there exist homogeneous subspaces \(X_n\) of \(X\) for \(n\in\omega\) such that \(X=\bigcup_{n\in\omega} X_n\). When each \(X_n\) is closed in \(X\) (respectively analytic or coanalytic in \(X\)), the space \(X\) is said to be \(\sigma\)-homogeneous with closed witnesses (respectively analytic witnesses or coanalytic witnesses). \textit{A. Ostrovsky} [Arch. Math. Logic 50, No. 5--6, 661--664 (2011; Zbl 1230.03078)] showed that \textit{every zero-dimensional Borel space is \(\sigma\)-homogeneous with pairwise disjoint closed witnesses.} The picture was further illustrated by results of \textit{A. Medini} and \textit{Z. Vidnyánszky} [Ann. Pure Appl. Logic 175, No. 1, Article ID 103331, 21 p. (2024; Zbl 1532.54019)], namely they proved that:\N\begin{itemize}\N\item \textit{Under AD, every zero-dimensional space is \(\sigma\)-homogeneous with pairwise disjoint closed witnesses.}\N\item \textit{In ZFC, there exists a zero-dimensional space that is not \(\sigma\)-homogeneous.}\N\item \textit{Assume V=L. Then there exists a coanalytic zero-dimensional space that is not \(\sigma\)-homogeneous.}\N\end{itemize}\NThe main result of the present paper is a theorem stating that \textit{every finite-dimensional analytic space is \(\sigma\)-homogeneous with analytic witnesses.} A disjoint variant of the main theorem asserts that \textit{every finite-dimensional analytic space is \(\sigma\)-homogeneous with pairwise disjoint \(\mathbf{\Delta}_2^1\) witnesses.} An example witnessing the optimality of the results is presented. The paper finishes with several open questions.