Abstract:A space $X$ is called $\mu $-compact by M. Mandelker if the intersection of all free maximal ideals of $C(X)$ coincides with the ring $C_K(X)$ of all functions in $C(X)$ with compact support. In this paper we introduce $\phi $-compact and $\phi '$-compact spaces and we show that a space is $\mu $-compact if and only if it is both $\phi $-compact and $\phi '$-compact. We also establish that every space $X$ admits a $\phi $-compactification and a $\phi '$-compactification. Examples and counterexamples are given.
Keywords: minimal prime ideal, $P$-space, $F$-space, $\mu $-compact space, $\phi $-compact space, $\phi '$-compact space, round subset, almost round subset, nearly round subset
AMS Subject Classification: Primary 54C40; Secondary 46E25