On countably generated $z$-ideals of $C(X)$ for first countable spaces
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- by Attilio Le Donne PDF
- Proc. Amer. Math. Soc. 82 (1981), 280-282 Request permission
Abstract:
In paper [L], a question asked in [D] has been answered first by proving that: if $X$ is normal and first countable, then every countably generated $z$-ideal of $C(X)$ is pure; then, by giving an example of a nonpure countably generated $z$-ideal of $C(X)$ in a $\sigma$-compact (hence normal) but not first countable space $X$. In this paper a class $\mathcal {C}$ of topological spaces $X$ whose $C(X)$ has a nonpure countably generated $z$-ideal is constructed; it is proved that $\mathcal {C}$ contains a space $X$ which is first countable. So it is proved that in the proposition above the hypotheses "normal" and "first countable" are both essential. Finally in $\S 4$ I prove, as announced in [L], that if $X$ is a locally compact normal space, then every countably generated $z$-ideal of $C(X)$ is pure. For the terminology and notations see [GJ], [D], [L].References
- G. De Marco, On the countably generated $z$-ideals of $C(X)$, Proc. Amer. Math. Soc. 31 (1972), 574–576. MR 288563, DOI 10.1090/S0002-9939-1972-0288563-3
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- Attilio Le Donne, On a question concerning countably generated $z$-ideals of $C(X)$, Proc. Amer. Math. Soc. 80 (1980), no. 3, 505–510. MR 581015, DOI 10.1090/S0002-9939-1980-0581015-7
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 280-282
- MSC: Primary 54C40; Secondary 54C50
- DOI: https://doi.org/10.1090/S0002-9939-1981-0609667-4
- MathSciNet review: 609667