Diagonals of separately continuous functions of n variables with values in strongly σ -metrizable spaces

Olena Karlova; Volodymyr Mykhaylyuk; Oleksandr Sobchuk

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 1, page 103-122
  • ISSN: 0010-2628

Abstract

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We prove the result on Baire classification of mappings f : X × Y → 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 P P -space, Y is a topological space and Z is a strongly σ -metrizable space with additional properties. We show that for any topological space X , special equiconnected space Z and a mapping g : X → Z of the ( n - 1 ) -th Baire class there exists a strongly separately continuous mapping f : X n → Z with the diagonal g . For wide classes of spaces X and Z we prove that diagonals of separately continuous mappings f : X n → 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 ] → Z , which is not a diagonal of any separately continuous mapping f : [ 0 , 1 ] 2 → Z , is constructed.

How to cite

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Karlova, Olena, Mykhaylyuk, Volodymyr, and Sobchuk, Oleksandr. "Diagonals of separately continuous functions of $n$ variables with values in strongly $\sigma $-metrizable spaces." Commentationes Mathematicae Universitatis Carolinae 57.1 (2016): 103-122. <http://eudml.org/doc/276820>.

@article{Karlova2016,
abstract = {We prove the result on Baire classification of mappings $f:X\times Y\rightarrow Z$ 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 $g:X\rightarrow Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\rightarrow Z$ with the diagonal $g$. For wide classes of spaces $X$ and $Z$ we prove that diagonals of separately continuous mappings $f:X^n\rightarrow Z$ are exactly the functions of the $(n-1)$-th Baire class. An example of equiconnected space $Z$ and a Baire-one mapping $g:[0,1]\rightarrow Z$, which is not a diagonal of any separately continuous mapping $f:[0,1]^2\rightarrow Z$, is constructed.},
author = {Karlova, Olena, Mykhaylyuk, Volodymyr, Sobchuk, Oleksandr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {diagonal of a mapping; separately continuous mapping; Baire-one mapping; equiconnected space; strongly $\sigma $-metrizable space},
language = {eng},
number = {1},
pages = {103-122},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Diagonals of separately continuous functions of $n$ variables with values in strongly $\sigma $-metrizable spaces},
url = {http://eudml.org/doc/276820},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Karlova, Olena
AU - Mykhaylyuk, Volodymyr
AU - Sobchuk, Oleksandr
TI - Diagonals of separately continuous functions of $n$ variables with values in strongly $\sigma $-metrizable spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 1
SP - 103
EP - 122
AB - We prove the result on Baire classification of mappings $f:X\times Y\rightarrow Z$ 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 $g:X\rightarrow Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\rightarrow Z$ with the diagonal $g$. For wide classes of spaces $X$ and $Z$ we prove that diagonals of separately continuous mappings $f:X^n\rightarrow Z$ are exactly the functions of the $(n-1)$-th Baire class. An example of equiconnected space $Z$ and a Baire-one mapping $g:[0,1]\rightarrow Z$, which is not a diagonal of any separately continuous mapping $f:[0,1]^2\rightarrow Z$, is constructed.
LA - eng
KW - diagonal of a mapping; separately continuous mapping; Baire-one mapping; equiconnected space; strongly $\sigma $-metrizable space
UR - http://eudml.org/doc/276820
ER -

References

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