A topological space X is called countably mesocompact if for every countably open cover U of X, there exists an open refinement V of U such that {V ∈ V : V ∩K 6= ∅} is finite for every compact set K in X. In this paper, we investigate relations between insertion of semi-continuous functions and a kind of spaces with countable mesocompactness , and give some characterizations of countably mesocompact spaces, k-perfect spaces and k-MCM spaces.