关于拓扑空间之间的散在连续映射
On scatteredly continuous maps between topological spaces
摘要：
拓扑空间之间的映射$f:X\to Y$被定义为{ \em散射连续}，如果对于每个子空间$A\subset X$，约束$f|A$有连续点.证明了对于从完全仿紧的遗传Baire Preiss-Simon空间$X$到正则空间$Y$的函数$f:X\to Y$，$f$的离散连续等价于（i）弱间断（对于每个子集$A\subset X$，$f|A$的不连续点的集合$D(f|A)$在$A$中没有密集的地方），（ii）$σ$ -连续性（$X$可以写为闭子集的可数并集，在闭子集上$f$是连续的），（iii）$G_δ$ -可测性（每个开集的原像是$G_δ$类型）。同样在Martin公理下，我们在可度量化可分空间之间构造了一个$G_δ$ -可测映射$f:X\to Y$，它不是分段连续的.这回答了维诺库罗夫的一个老问题。
英文摘要：
A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact hereditarily Baire Preiss-Simon space $X$ into a regular space $Y$ the scattered continuity of $f$ is equivalent to (i) the weak discontinuity (for each subset $A\subset X$ the set $D(f|A)$ of discontinuity points of $f|A$ is nowhere dense in $A$), (ii) the $σ$-continuity ($X$ can be written as a countable union of closed subsets on which $f$ is continuous), (iii) the $G_δ$-measurability (the preimage of each open set is of type $G_δ$). Also under Martin Axiom, we construct a $G_δ$-measurable map $f:X\to Y$ between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov.
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