One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces
We investigate the existence of a separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with a one-point set of points of discontinuity in the case where the topological spaces $X$ and $Y$ satisfy conditions of compactness type. In particular, for the compact spaces $X$ and $Y$ a...
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| Date: | 2005 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2005
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3576 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We investigate the existence of a separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with a one-point set of points of discontinuity in the case where the topological spaces $X$ and $Y$ satisfy conditions of compactness type.
In particular, for the compact spaces $X$ and $Y$ and the nonizolated points $x_0 \in X$ and $y_0 \in Y$, we show that the separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with the set of points of discontinuity $\{(x_0, y_0)\}$ exists if and only if sequences of nonempty functionally open set exist in $X$ and $Y$ and converge to $x_0$ and $y_0$, respectively. |
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