On the countably-compactifiability in the sense of Morita

We consider an extension $Y$ of a topological space $X$ that is canonically embedded in the Wallman extension $\omega X$, in which any countably compact set closed in $X$ is closed and such that any infinite set contained in $X$ has a limit point in it. This extension is called saturation of the space $X$. We find a necessary and sufficient condition for the countable compactness of the space $Y$. Thus the problem of existence of countably-compactification in the sense of Morita of certain type is solved.