On the sequential order continuity of the $C(K)$-space

As shown in [1], for each compact Hausdorff space $K$ without isolated points, there exists a compact Hausdorff $P'$-space $X$ but not an $F$-space such that $C(K)$ is isometrically Riesz isomorphic to a Riesz subspace of $C(X)$. The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.