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Proceedings of the American Mathematical Society

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On countably generated $ z$-ideals of $ C(X)$ for first countable spaces


Author: Attilio Le Donne
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
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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 $ \S4$ 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].


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DOI: https://doi.org/10.1090/S0002-9939-1981-0609667-4
Article copyright: © Copyright 1981 American Mathematical Society