## 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 $$I^{\tau}$$.

### MSC:

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