Diagonals of separately continuous functions of $n$ variables with values in strongly $\sigma$-metrizable spaces

Karlova, Olena ; Mykhaylyuk, Volodymyr ; Sobchuk, Oleksandr

Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 1, pp. 103-122.

Résumé

We prove the result on Baire classification of mappings $f:X\times Y\to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $PP$-space, $Y$ is a topological space and $Z$ is a strongly $\sigma$-metrizable space with additional properties. We show that for any topological space $X$, special equiconnected space $Z$ and a mapping $g:X\to Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\to Z$ with the diagonal $g$. For wide classes of spaces $X$ and $Z$ we prove that diagonals of separately continuous mappings $f:X^n\to Z$ are exactly the functions of the $(n-1)$-th Baire class. An example of equiconnected space $Z$ and a Baire-one mapping $g:[0,1]\to Z$, which is not a diagonal of any separately continuous mapping $f:[0,1]^2\to Z$, is constructed.

Détail

DOI : 10.14712/1213-7243.2015.152

Classification : 26B05, 54C05, 54C08
Keywords: diagonal of a mapping; separately continuous mapping; Baire-one mapping; equiconnected space; strongly $\sigma$-metrizable space