Sequential closure of the space of jointly continuous functions in the space of separately continuous functions
Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$, and $|| f||_y = \mathrm{max}_{x \...
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| Date: | 2016 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2016
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1830 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$,
and
$|| f||_y = \mathrm{max}_{x \in X} |f(x, y)|,\; y \in Y$.
Under the assumption that the compact space $X$ is metrizable, we prove that a
separately continuous function $f : X \times Y \rightarrow R$ is the limit of a sequence $(f_n)^{\infty}_{n=1}$ of jointly continuous function $f_n : X \times Y \rightarrow R$ in $S(X \times Y )$ provided that the set $D(f)$ of discontinuity points of $f$ has countable projections on $X$. |
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