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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
Zbl 1152.54331
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