On Baire class one functions on a product space
Let \(X\) be a Tychonoff product of a countable collection of non-singleton completely regular spaces \(X_j\) and let \(S\) be a subset of \(X\). A function \(f:S\to\mathbb{R}\) is said to be finitely determined if there exists \(m\in\mathbb{N}\) such that \(f(x)=f(y)\) for all \(x,y\in S\) whose first \(m\) coordinates coincide. Let \(CF(S)\) be the set of all continuous finitely determined functions from \(S\) to \(\mathbb{R}\). The author shows that every continuous function \(f:X\to\mathbb{R}\) is the uniform limit of a sequence in \(CF(X)\) if and only if \(X\) is pseudocompact. Further, if \(X\) is metrizable, then every Baire class \(\alpha\) function \(f:S\to\mathbb{R}\) is the pointwise limit of a sequence of finitely determined functions in previous Baire classes if and only if \(\alpha=1\). Moreover, a lower semicontinuous function on a metrizable \(X\) is the pointwise limit of an increasing sequence in \(CF(X)\) if and only if all but finitely many of \(X_j\) are compact.
scientific article; zbMATH DE number 6534246