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A space $X$ is called $\mu$-compact by M. Mandelker if the intersection of all free maximal ideals of $C\left(X\right)$ coincides with the ring $C_K\left(X\right)$ of all functions in $C\left(X\right)$ with compact support. In this paper we introduce $\phi$-compact and ${\phi }^{\text{'}}$-compact spaces and we show that a space is $\mu$-compact if and only if it is both $\phi$-compact and ${\phi }^{\text{'}}$-compact. We also establish that every space $X$ admits a $\phi$-compactification and a ${\phi }^{\text{'}}$-compactification. Examples and counterexamples are given.