  The self-closeness number of a CW-complex is a homotopy invariant defined by
the minimal number $n$ such that every self-maps of $X$ which induces
automorphisms on the first $n$ homotopy groups of $X$ is a homotopy
equivalence. In this article we study the self-closeness numbers of finite
Cartesian products, and prove that under certain conditions (called
reducibility), the self-closeness number of product spaces equals to the
maximum of self-closeness numbers of the factors. A series of criteria for the
reducibility are investigated, and the results are used to determine
self-closeness numbers of product spaces of some special spaces, such as Moore
spaces, Eilenberg-MacLane spaces or atomic spaces.
