  The asymptotic behavior for fully coupled multiscale stochastic systems
becomes much complicated when the fast processes do not locate in a compact
space. An illustrative example is constructed, which shows that the averaged
coefficients may become discontinuous even they are originally Lipschitz
continuous. We aim to reveal the impact of the ergodicity of the fast process
on the limit process and the averaging principle. The crucial point is the
continuous dependence on parameters related to the slow process for the
invariant probability measures in various distances over the Wasserstein space.
