Baire isomorphisms at the first level and dimension

For a topological space X let Zσ(X) denote the family of subsets of X which can be represented as a union of countably many zero-sets. A bijection h :X→Y between topological spaces X and Y is a first level Baire isomorphism if f(Z)∈Zσ(Y) and f−1(Z′)∈Zσ(X) whenever Z∈Zσ(X) and Z′∈Zσ(Y). A space is σ-(pseudo)compact if it can be represented as the union of a countable family consisting of its (pseudo)compact subsets. Generalizing results of Jayne, Rogers and Chigogidze we show that first level Baire isomorphic, σ-pseudocompact (in particular, σ-compact) Tychonoff spaces have the same covering dimension dim.