J. E. Jayne

σ-fragmentable Banach spaces

§1. Introduction . Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X . The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ -diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each e>0, it is possible to write where each set X i , i ≥1, has the property that each non-empty subset of X i , has a non-empty relatively open subset of ρ -diameter less than e. If is any family of subsets of X , we say that X is σ-fragmented by the metric ρ , using sets from , if, for each e>0, the sets X i , i ≥ 1, in (1.1) can be taken from

Borel isomorphisms at the first level—I

A Borel isomorphism that, together with its inverse, maps ℱ σ -sets to ℱ σ -sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism , for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.

Structure of analytic Hausdorff spaces

In Hausdorff topological spaces there are currently three definitions of analytic sets due respectively to Choquet [1], Sion [8], and Frolīk [3, 4]. Here it is shown that these definitions are equivalent.

Continuous functions on products of compact Hausdorff spaces

In [4], we investigated the spaces of continuous functions on countable products of compact Hausdorff spaces. Our main object here is to extend the discussion to arbitrary products of compact Hausdorff spaces. We prove the following theorems in Section 3.

Piece-wise closed functions and almost discretely σ-decomposable families

In [8,9] Jayne and Rogers studied piece-wise closed maps and ℱ σ maps between metric spaces. A map f of a metric space X into a metric space Y is said to be an ℱ σ map if: (a) f maps ℱ σ -sets in X to ℱ σ -sets in Y ; and (b) f 1 maps ℱ σ -sets in Y back to ℱ σ -sets in X . A map f of a metric space X into a metric space Y is said to be piece-wise closed if:it is possible to find a sequence X 1 , X 2 ,… of closed sets in X , with with each set f ( X i ), i ≥ 1, closed in Y , and with the restriction of f to each X i , a closed map (i.e., a continuous map that maps closed sets to closed sets).

Borel isomorphisms at the first level: Corrigenda et addenda

In recent work [2] we have investigated Borel isomorphisms at the first level, i.e. mappings that together with their inverses map F σ -sets to F σ -sets. We are grateful to Dr. F. Topsoe for writing to point out errors in the proofs of our Theorems 2.1, 2.2 and 2.3. The errors escaped our attention because we only wrote out the first step of a complicated inductive argument and carelessly and falsely claimed that the general step of the induction was similar to the first. To be more specific, in the proof of Theorem 2.1, although we do have we do not, in general, have Fortunately the proofs can be corrected.

K -analytic sets: corrigenda et addenda

In [4] we initiated a study of K -Lusin sets. We characterized the K -Lusin sets in a Hausdorff space X as the sets that can be obtained as the image of some paracompact Cech complete space G , under a continuous injective map that maps discrete families in G to discretely σ -decomposable families in X [4, Theorem 2, p. 195]. Unfortunately, we cannot substantiate a second characterization of K -Lusin sets in completely regular spaces, given in the second part of Theorem 14 of [4].