Function Characterizations of Countably Paracompact Space and Countably Mesocompact

Some characterizations of countably paracompact spaces and countably mesocompact spaces, were given with semi-continuous funtions. The main results are as follows. X is a countably paracompact if and only if for every decreasing sequences g_n ∈ L(X):n ∈ N and h_n ∈ U(X):n ∈ N of functions such that for each n ∈ N f_n ≤ g_n ≤ h_n and h_n→0. X is a countably mesocompact if and only if for each upper semi-continuous function f, there exists a lower semi-continuous and k-upper bounded function ϕ(f), such that f ≤ ϕ(f).