Discontinuity points of separately continuous mappings with values in the Sorgenfrey line (in Ukrainian)

Author
V. K. Maslyuchenko, O. D. Myronyk
Chernivtsi National University
Abstract
We prove the following results. Let $X$ and $Y$ be topological spaces with $Y$ first countable. If $f\colon X\times Y\to \mathbb L$ is a separately continuous mapping with values in the Sorgenfrey line $\mathbb L$ then for every $y\in Y$ the set $D_y(f)=\{x\in X\colon (x,y)$ is a discontinuity point of $f$ is meager in $X$. Conversely, for every meager $F_\sigma$-set $A$ in $\mathbb L$ and for arbitrary point $b$ of the rational line $\mathbb Q$ we construct the separately continuous mapping $f\colon \mathbb L\times\mathbb Q\to \mathbb L$, such that the set of all discontinuity points is equal to $A\times \{b\}$.
Keywords
Sorgenfrey line; separately continuous mapping
DOI
doi:10.15330/ms.45.1.67-75
Reference
1. Filipchuk O.I. Separately continuous mappings and their analogues with values in non-metrizable spaces. - PhD thesis, 2010. (in Ukrainian)

2. Myronyk O.D. Stratifiable, semistratifiable spaces and separately continuous mappings. - PhD thesis, 2015. (in Ukrainian)

3. Maslyuchenko V.K., Filipchuk O.I. The importance of the $\sigma$-metrizability in results on joint continuity of KC-functions// Int. Conf. dedicate to the 125th anniversary of Hans Hahn. Book of Abstracts. - 2004. - P. 65-66. (in Ukrainian)

4. Maslyuchenko V.K., Myronyk O.D. The Sorgenfrey line and separately continuous mappings, Buk. Math. J. - 2014. - V.2, 1. - P. 59-68. (in Ukrainian)

5. Breckenridge J.C., Nishiura T. Partial continuity, quasicontinuity and Baire spaces// Bull. Inst. Acad. Sinica. - 1976. - V.4, 2. - P. 191-203.

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Pages
67-75
Volume
45
Issue
1
Year
2016
Journal
Matematychni Studii
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