o-homogeneity of analytic spaces

Determine whether every analytic separable metrizable topological space is o-homogeneous; that is, ascertain whether for every analytic space X there exist homogeneous subspaces X_n of X for n in ω such that X = ⋃_{n∈ω} X_n.

Background

The paper proves that every finite-dimensional analytic space is o-homogeneous, extending prior results from the zero-dimensional setting. The authors also obtain optimal complexity bounds for the witnessing subspaces and show pairwise disjoint witnesses exist with higher (optimal) complexity. Despite completing the finite-dimensional picture, the general analytic case remains unresolved.

Throughout, “space” means a separable metrizable topological space, and “analytic” is in the descriptive-set-theoretic sense. o-homogeneity asks for a countable union of homogeneous subspaces covering the space, with attention to the descriptive complexity of the witnesses.