On countably generated -ideals of for first countable spaces
Author: Attilio Le Donne
Journal: Proc. Amer. Math. Soc. 82 (1981), 280-282
MSC: Primary 54C40; Secondary 54C50
MathSciNet review: 609667
Abstract: In paper [L], a question asked in [D] has been answered first by proving that: if is normal and first countable, then every countably generated -ideal of is pure; then, by giving an example of a nonpure countably generated -ideal of in a -compact (hence normal) but not first countable space .
In this paper a class of topological spaces whose has a nonpure countably generated -ideal is constructed; it is proved that contains a space which is first countable. So it is proved that in the proposition above the hypotheses "normal" and "first countable" are both essential.
Finally in I prove, as announced in [L], that if is a locally compact normal space, then every countably generated -ideal of is pure.
For the terminology and notations see [GJ], [D], [L].
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- [GJ] 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
- [L] 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, https://doi.org/10.1090/S0002-9939-1980-0581015-7
- G. De Marco, On the countably generated -ideals of , Proc. Amer. Math. Soc. 31 (1972), 574-576. MR 0288563 (44:5760)
- L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, New York, 1960. MR 0116199 (22:6994)
- A. Le Donne, On a question concerning countably generated -ideals of , Proc. Amer. Math. Soc. 80 (1980), 505-510. MR 581015 (81j:54014)