两个紧空间乘积上分别连续函数的单点间断集
Onepoint discontinuity set of separately continuous functions on the product of two compact spaces
摘要：
研究了满足紧型条件的拓扑空间$X$和$Y$中具有单点间断集的分离连续函数$f:X\times Y\to \mathbb R$的存在性.特别地，证明了对于紧空间$X$和$Y$以及非隔离点$x_0\in X$和$y_0\in Y$，存在一个具有不连续点集合$\{(x_0,y_0)\}$的独立连续函数$f:X\times Y\to \mathbb R$当且仅当存在分别收敛于$X$和$Y$中的$x_0$和$y_0$的非空泛函开集序列.
英文摘要：
It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown that for compact spaces $X$ and $Y$ and nonizolated points $x_0\in X$ and $y_0\in Y$ there exists a separately continuous function $f:X\times Y\to \mathbb R$ with the set $\{(x_0,y_0)\}$ of discontinuity points if and only if there exist sequences of nonempty functional open sets which converge to $x_0$ and $y_0$ in $X$ and $Y$ respectively.
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