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

Karamzadeh, O. A. S.; Rostami, M.

doi:10.1090/S0002-9939-1985-0766552-9

On the intrinsic topology and some related ideals of $C(X)$
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by O. A. S. Karamzadeh and M. Rostami

Proc. Amer. Math. Soc. 93 (1985), 179-184

DOI: https://doi.org/10.1090/S0002-9939-1985-0766552-9

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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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