Every finite-dimensional analytic space is $σ$-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 $σ$-homogeneous with analytic witnesses, (2) Every finite-dimensional analytic space is $σ$-homogeneous with pairwise disjoint $\mathbfΔ^1_2$ witnesses. Furthermore, the complexity of the witnesses is optimal in both of the above results. This completes the picture regarding $σ$-homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is $σ$-homogeneous. We also investigate finite unions of homogeneous spaces.

Theorems & Definitions (28)

• Theorem 1.1: Ostrovsky
• Theorem 1.2: Medini, Vidnyánszky
• Theorem 1.3: Medini, Vidnyánszky
• Theorem 1.4: Medini, Vidnyánszky
• Lemma 2.1
• proof
• Theorem 3.1: van Engelen
• Lemma 3.2: Harrington
• Lemma 3.3
• proof