SOME CHARACTERIZATIONS OF SINGULAR COMPACTIFICATIONS

SOME CHARACTERIZATIONS OF SINGULAR COMPACTIFICATIONS

Assume that X is locally compact and Hausdorff. Then, we show that $\alpha X = sup {X \cup_f S(f)$\mid$f \in S^{\alpha}}$ 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 $+\infty \notin A$. As a corollary, we obtain that if X is connected and locally connected, then $\alpha X = sup {X \cup_f S(f)$\mid$f \in S^{\alpha}}$ for any compactification $\alpha X$ of X if and only if X is 1-complemented.

(0)

(0)

(0)

(0)