Absolute Souslin-F spaces and other weak-invariants of the norm topology
✍ Scribed by Roger W. Hansell
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 101 KB
- Volume
- 111
- Category
- Article
- ISSN
- 0166-8641
No coin nor oath required. For personal study only.
✦ Synopsis
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 F σ -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 ℵ 1 and being a σ -discrete set are weak-invariants of the norm topology. Other weak-invariants such as (A, weak) being σfragmented by the norm are also established.