A characterization of $N$-compact spaces

In this paper, we prove the following theorem: Theorem A. A $0$-dimensional space $X$ is $N$-compact if and only if every clopen ultrafilter on $X$ with the countable intersection property is fixed, where $N$ is the space of all natural numbers. Two consequences of Theorem A are as follows: Theorem B. Suppose that $X$ and $Y$ are $N$-compact spaces. A mapping $\phi$ from the Boolean ring $\mathfrak {B}(X)$ of all clopen subsets of $X$ onto the Boolean ring $\mathfrak {B}(X)$ of all clopen subsets of $Y$ is an isomorphism with the property that $\bigcap \nolimits _{i = 1}^\infty {{A_i} = \emptyset ({A_i} \in \mathfrak {B}(X))}$ implies $\bigcap \nolimits _{i = 1}^\infty \phi ({A_i}) = \emptyset$ if and only if there exists a homeomorphism $h$ from $X$ onto $Y$ such that $\phi (A) = h[A]$ for each $A$ in $\mathfrak {B}(X)$. Theorem C. A $0$-dimensional space $X$ is $N$-compact if and only if the collection of all the countable clopen coverings of $X$ is complete.