中文摘要：

首先,借助和拓扑,对于离散拓扑空间X,证明了任意一族以X为指标集的拓扑空间族的积空间和相对于X的点式收敛空间相同。其次,对于某拓扑空间族的笛卡尔积和其上的任意一个拓扑,给出了该积空间到其任意坐标空间的投射连续的充要条件。最后,给出了函数式积空间的定义,并且指出这类空间可以作为积空间、和空间和函数空间（点式收敛拓扑）的共同推广。

英文摘要：

Based on the topological sum,for Xbeing a discrete topological space,we prove that the product topology generated by some topological spaces is equal to the topology of pointwise convergence related to X.For every topology on a Cartesian product,we find an equivalent condition under which every projection from the product to each component is continuous.We propose the definition of functional product spaces,which can be considered as a common framework of the product spaces,the topological sums and the functional spaces（the topology of pointwise convergence）.