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Diagonals of separately continuous functions of \(n\) variables with values in strongly \(\sigma \)-metrizable spaces. (English) Zbl 1374.54019
For a function \(f:X^n\to Y\) the mapping \(g:X\to Y\) defined by \(g(x)=f(x,\dots, x)\) is called a diagonal of \(f\). It is well-known that for \(n\geq 2\) diagonals of separately continuous functions \(f:\mathbb{R}^n\to\mathbb{R}\) are exactly the functions of the \((n-1)\)-th Baire class. If \(X\) is any topological space then for every function \(g:X\to\mathbb{R}\) of the \(n\)-th Baire class there exists a separately continuous function \(f:X^{n+1}\to\mathbb{R}\) with the diagonal \(g\), see V. V. Mykhaĭlyuk [Ukr. Mat. Visn. 3, No. 3, 374–381 (2006); translation in Ukr. Math. Bull. 3, No. 3, 361–368 (2006; Zbl. 1152.54331)]. In the paper under review the authors study the analogous problem for functions with values in a space \(Z\) from a wide class of spaces which contains metrizable equiconnected spaces and strict inductive limits of sequences of closed locally convex metrizable subspaces. Namely, assuming that \(X\) is a topological space and \(Z\) is a strongly \(\sigma\)-metrizable equiconnected space with a perfect stratification they prove that for a given function of the \(n\)-th Baire class \(g:X\to Z\) there exists a separately continuous function \(f:X^{n+1}\to Z\) with the diagonal \(g\). They also construct an example of an equiconnected space \(Z\) and a Baire-one function \(g:[0,1]\to Z\) which is not a diagonal of any separately continuous function \(f:[0,1]^2\to Z\).

MSC:
54C08 Weak and generalized continuity
54C05 Continuous maps
26B05 Continuity and differentiation questions
Citations:
Zbl 1152.54331
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