Assume that X is locally compact and Hausdorff. Then, we show that $alpha X = sup {X cup_f S(f)$mid$f in S^{alpha}}$</TEX> for any compactification $alpha X$ of X if and only if for any 2-point compatification $gamma X$ of X with $gamma X - X = {-infty, +infty}$, there exists a clopen subset A of gamma X$ such that $-infty in A$ and $+infty otin A$. As a corollary, we obtain that if X is connected and locally connected, then $alpha X = sup {X cup_f S(f)$mid$f in S^{alpha}}$</TEX> for any compactification $alpha X$ of X if and only if X is 1-complemented.