On Baire isomorphisms of spaces of uncountable weight. (English. Russian original) Zbl 0632.54012

Sov. Math., Dokl. 32, No. 1, 113-117 (1985); translation from Dokl. Akad. Nauk SSSR 283, No. 2, 321-325 (1985).
For a completely regular space X let \(B_ 0(X)\) denote the \(\sigma\)- algebra of Baire subsets of X. A space X is called an absolute Baire space if \(X\in B_ 0(\beta X)\). A mapping \(f: X\to Y\) is said to be \(B_ 0\)-measurable if \(f^{-1}(B_ 0(Y))\subset B_ 0(X)\). A one-to- one mapping \(f: X\to Y\) of a space X onto a space Y is called a Baire isomorphism (briefly, a \(B_ 0\)-isomorphism) if f and \(f^{-1}\) are \(B_ 0\)-measurable mappings. We study the question of existence of a Baire isomorphism between Tikhonov cubes, on the one hand, and subsets of Dugundji and \(\kappa\)-metrizable compact Hausdorff spaces, on the other hand. The following theorem is the main result: Theorem. If X is homogeneous with respect to character and is a Baire subset of Dugundji space, then it is Baire isomorphic to \(I^{\omega (X)}\). Corollary. A Dugundji space of weight \(\tau\) that is homogeneous with respect to character is Baire isomorphic to \(I^{\tau}\).


54C50 Topology of special sets defined by functions
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54E52 Baire category, Baire spaces