On Baire classification of strongly separately continuous functions.

*(English)*Zbl 1390.54015The author considers strongly separately continuous functions defined on a subspace of a product of topological spaces \(\{X_t\}_{t\in T}\) equipped with the Tychonoff topology of pointwise convergence. Let \(X=\prod_{t\in T} X_t\), \(E\subset X\), \(a\in E\), and \(t\in T\). For \(x\in X\) define \(x_t^a:T\to\mathbb{R}\) by \(x^a_t(i)=x(i)\) for \(i\neq t\), and \(x^a_t(t)=a(t)\). A function \(f:E\to\mathbb{R}\) is strongly separately continuous at a point \(a\) with respect to the \(t\)-th variable if \(\lim_{x\to a} |f(x)-f(x^a_t)|=0\). A mapping \(f:E\to\mathbb{R}\) is strongly separately continuous at a point \(a\in E\) if \(f\) is strongly separately continuous at \(a\) with respect to each variable \(t \in T\), and \(f\) is strongly separately continuous on the set \(E\) if it is strongly separately continuous at every point \(a \in E\) with respect to each variable \(t\in T\). Recall that for \(f:\mathbb{R}^n\to\mathbb{R}\), \(n\in\mathbb{N}\), the properties of strong separate continuity and continuity are equivalent, see O. Dzagnidze [Real Anal. Exch. 24, No. 2, 695–702 (1999; Zbl 0967.26010)]. The main results of the paper under review are:

(1) If \(X=\prod_{n\in\mathbb{N}}X_n\) is a product of countably many topological spaces, \(a\in X\) and \(\sigma(a)\) is the set of all \(x\in X\) with \(|\{ n: x_n\neq a_n\}|<\aleph_0\), then every strongly separately continuous function \(f:\sigma(a)\to\mathbb{R}\) belongs to the first stable Baire class, i.e. there exists a sequence \(f_i:\sigma(a)\to\mathbb{R}\) of continuous functions such that for every \(x\in\sigma(a)\) there is \(i_x\in\mathbb{N}\) with \(f_i(x)=f(x)\) for \(i\geq i_x\).

(2) If \(X\) is a product of countably many real lines, then there exists a strongly separately continuous function \(f:X\to\mathbb{R}\) which is not Baire measurable.

(3) If \(X\) is a countable product of normed spaces, then for any open set \(G\subset\sigma(a)\) there is a strongly separately continuous function \(f:\sigma(a)\to\mathbb{R}\) such that \(G\) is equal to the set of all points \(x\in \sigma(a)\) at which \(f\) is discontinuous.

(1) If \(X=\prod_{n\in\mathbb{N}}X_n\) is a product of countably many topological spaces, \(a\in X\) and \(\sigma(a)\) is the set of all \(x\in X\) with \(|\{ n: x_n\neq a_n\}|<\aleph_0\), then every strongly separately continuous function \(f:\sigma(a)\to\mathbb{R}\) belongs to the first stable Baire class, i.e. there exists a sequence \(f_i:\sigma(a)\to\mathbb{R}\) of continuous functions such that for every \(x\in\sigma(a)\) there is \(i_x\in\mathbb{N}\) with \(f_i(x)=f(x)\) for \(i\geq i_x\).

(2) If \(X\) is a product of countably many real lines, then there exists a strongly separately continuous function \(f:X\to\mathbb{R}\) which is not Baire measurable.

(3) If \(X\) is a countable product of normed spaces, then for any open set \(G\subset\sigma(a)\) there is a strongly separately continuous function \(f:\sigma(a)\to\mathbb{R}\) such that \(G\) is equal to the set of all points \(x\in \sigma(a)\) at which \(f\) is discontinuous.

Reviewer: Tomasz Natkaniec (Gdańsk)