Abstract
In this paper, we make several observations on cellular-Lindelöf spaces. We prove that in perfect spaces, the property of being cellular-Lindelöf is equivalent to the countable chain condition. Using this result, we prove that every cellular-Lindelöf first-countable perfect space has cardinality at most \({\mathfrak {c}}\) and obtain a regular example of a weakly Lindelöf non-cellular-Lindelöf space. We also prove that if X is a cellular-Lindelöf space then every discrete family of non-empty open subsets of X is countable. Finally, we prove that if X is a cellular-Lindelöf space with a symmetric g-function such that \(\cap \{g^2(n,x): n\in \omega \}=\{x\}\) for each \(x\in X\) then \(|X| \le 2^{ {\mathfrak {c}}}\).
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Acknowledgements
This research was supported by supported by NSFC project 11626131 and 11771029. The authors also would like to thank the referee for his (or her) valuable remarks and suggestions which greatly improved the paper.
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Communicated by Mohammad Reza Koushesh.
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Xuan, WF., Song, YK. On Cellular-Lindelöf Spaces. Bull. Iran. Math. Soc. 44, 1485–1491 (2018). https://doi.org/10.1007/s41980-018-0102-1
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DOI: https://doi.org/10.1007/s41980-018-0102-1