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 δH : Ƒ(X)² → [0, ∞] 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(Δ) 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.

Keywords: Effros-Borel structure, Vietoris topology, Hausdorff metric, spherical opening, geometric opening, ball opening, Banach-Mazur, operator opening.