# Article

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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
Summary:
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.
References:
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