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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
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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[2] Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110-130. MR 0194880 | Zbl 0147.29105
[3] Johnson D.G., Mandelker M.: Functions with pseudocompact support. General Topology Appl. 3 (1973), 331-338. MR 0331310 | Zbl 0277.54009
[4] Mandelker M.: Round $z$-filters and round subsets of $\beta X$. Israel J. Math. 7 (1969), 1-8. MR 0244951 | Zbl 0174.25604
[5] Mandelker M.: Supports of continuous functions. Trans. Amer. Math. Soc. 156 (1971), 73-83. MR 0275367 | Zbl 0197.48703
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