Notes on selectively 2-star-ccc spaces

Remarks on set selectively star-CCC spaces

“…n∈N + A n , U , as required. In [11], Xuan and Song showed that CCC implies selectively 2-starccc, which gives a positive answer to Problem 4.4 in [7]. Now we can obtain their result by Corollary 13.…”

“…The results in this paper follow directly from work in [3,11]. For completeness, we present proofs in our current terminology.…”

“…By the Corollary 2 above, we could give a negative answer to a Problem raised by Xuan and Song ([11], Question 4.10); Since selectively 3-star-ccc equals weakly star-Lindelöf for normal space, there does not exist a selectively 3-star-ccc normal space which is not weakly star-Lindelöf. Note that weakly star-Lindelöf is called weakly star countable in [11].…”

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In this paper, the author represent a unification and extension of concepts previously studied by several authors. By establish connections between the chain condition, selectively starccc properties and star-Lindelöf properties, the author provide that selectively 3-star-ccc equals weakly strongly star-Lindelöf for normal spaces, which gives a negative answer to Question 4.10 raised by Xuan and Song in [11], and use a different method from Xuan and Song to provide that the CCC implies selectively 2-star-ccc, which gives a positive answer to Problem 4.4 in [7]. Finally, an example is given to distinguish these properties.

1.IntroductionA systematic study of classical selection principles was started by Scheepers in [4].

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“…Recently, Song and Xuan investigated the equivalence and properties of certain of these spaces. Several questions were raised in [11]. Most of these questions remains open.This paper arose in the attempt to analyse these questions.…”