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Proceedings of the American Mathematical Society

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On the intrinsic topology and some related ideals of $ C(X)$

Authors: O. A. S. Karamzadeh and M. Rostami
Journal: Proc. Amer. Math. Soc. 93 (1985), 179-184
MSC: Primary 54C40
MathSciNet review: 766552
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Abstract: The above topology is defined and studied on $ C(X)$, the ring of real-valued continuous functions on a completely regular Hausdorff space $ X$. The minimal ideals and the socle of $ C(X)$ are characterized via their corresponding $ z$-filters. We observe that these ideals are $ z$-ideals and $ X$ is discrete if and only if the socle of $ C(X)$ is a free ideal. It is also shown that for a class of topological spaces, containing all $ P$-spaces, the family $ {C_k}(X)$ of functions with compact support is identical with the intersection of the free maximal ideals of $ C(X)$.

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Keywords: Isolated maximal ideals, intrinsic topology, $ P$-space, minimal ideal, socle, real pseudo-finite space, $ P$-ideal, $ z$-ideal
Article copyright: © Copyright 1985 American Mathematical Society