Absolute Souslin-F spaces and other weak-invariants of the norm topology

Let h :(A,weak)→(B,weak) be a homeomorphism where A and B are arbitrary subsets of (possibly different) Banach spaces. Then any property that holds for (B,norm) whenever it holds for (A,norm) is said to be a weak-invariant of the norm topology. We show that, relative to the norm topologies on A and B, the map h and its inverse are Fσ-measurable and take norm discrete collections to norm σ-discretely decomposable collections. We deduce from this a number of properties that are weak-invariants of the norm topology, including such properties as being an absolute Borel space, being an absolute Souslin-F space, and being σ-locally of weight less than some infinite cardinal κ. The latter two properties generalize results of Namioka and Pol (1993) who showed previously that being an absolute Souslin-F space of weight ⩽ℵ1 and being a σ-discrete set are weak-invariants of the norm topology. Other weak-invariants such as (A,weak) being σ-fragmented by the norm are also established.