On the Hausdorff distance and some openings between Banach spaces as Borel functions

Abstract

In these notes we show that the infinite valued Hausdorff distance δ_H: F (X)² → [0, 8] is Borel. Also, it is shown that the spherical opening, the geometric opening and the ball opening are Borel functions from SB² to [0, 1], where SB stands for the standard coding of separable Banach spaces as closed subspaces of C (∆) endowed with the Effros-Borel structure. Also, we discuss the Borel complexity of the Banach-Mazur distance, and we show that its restriction to finite dimensional Banach spaces is Borel.