Abstract
We show that the component and quasi component of 0 in the space C r (X) (C(X) with r-topology), coincide with the intersection of all principal ideals in C*(X) generated by regular elements of C*(X). Several different characterizations of the component of 0 in C r (X) are given and using these characterizations, it turns out that C r (X) is totally disconnected if and only if the set of non-almost P-points of β X is dense in β X. It is also shown that C r (X) is connected or locally connected if and only if X is a pseudocompact almost P-space. We observe that most of the familiar forms of compactness and most of the countability properties of C r (X) are equivalent to the finiteness of X. Finally, open and closed ideals in C r (X) are investigated and it is proved that for every \({p \in X}\), cl r O p = M p if and only if X is an almost P-space.
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Azarpanah, F., Paimann, M. & Salehi, A.R. Compactness, connectedness and Countability properties of C(X) with the r-topology. Acta Math. Hungar. 146, 265–284 (2015). https://doi.org/10.1007/s10474-015-0509-6
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DOI: https://doi.org/10.1007/s10474-015-0509-6
Key words and phrases
- r-topology
- hemicompact
- \({\sigma}\)-compact
- component
- quasi component
- totally disconnected and almost P-space