In this article, we will study the equivariant cohomology theory for actions of compact group (not necessarily Lie group) on compact spaces. We will present a somewhat more general and shorter proof of the localization theorem, known as the Borel-Hsiang-Quillen localization theorem, which was generalized by Özkurt and Onat to actions of finite-dimensional compact groups on compact connected spaces. In particular, we will apply this to the problem of the existence of equivariant maps between topological transformation groups.