Geodesic
Intersections of minimal prime ideals in the rings of continuous functions
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 4, pp. 623-632.

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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.
Classification : 46E25, 46J20, 54C40
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 
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Ghosh, Swapan Kumar. Intersections of minimal prime ideals in the rings of continuous functions. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 4, pp. 623-632. http://geodesic.mathdoc.fr/item/CMUC_2006__47_4_a7/