Every finite-dimensional analytic space is $\sigma$-homogeneous

Abstract

All spaces are assumed to be separable and metrizable. Building on work of van Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following results: (1) Every finite-dimensional analytic space is $\sigma$-homogeneous with analytic witnesses, (2) Every finite-dimensional analytic space is $\sigma$-homogeneous with pairwise disjoint ${\mathbf{\Delta }}_2^1$ witnesses. Furthermore, the complexity of the witnesses is optimal in both of the above results. This completes the picture regarding $\sigma$-homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is $\sigma$-homogeneous. We also investigate finite unions of homogeneous spaces.

10 pages. arXiv admin note: text overlap with arXiv:2107.07747