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- Author or Editor: Yan-Kui Song x

- Mathematics and Statistics x

A space *X* is *almost star countable (weakly star countable)* if for each open cover *U* of *X* there exists a countable subset *F* of *X* such that

## Summary

A subset *Y* of a space *X* is weakly almost Lindelf in *X* if for every open cover U of *X*, there exists a countable subfamily V of U such that *Y* ⊆ comp (∩V ). We investigate the relationship between relatively weakly almost Lindelf subsets and relatively almost Lindelf
subsets, and also study various properties of relatively weakly almost Lindelf subsets.

## Abstract

In this paper, we prove that if *X* is a space with a regular *G*_{δ}-diagonal and *X*^{2} is star Lindelöf then the cardinality of *X* is at most 2^{c}. We also prove that if *X* is a star Lindelöf space with a symmetric *g*-function such that *g*^{2}(*n, x*): *n* ∈ *ω*} = {*x*} for each *x* ∈ *X* then the cardinality of *X* is at most 2^{c}. Moreover, we prove that if *X* is a star Lindelöf Hausdorff space satisfying *Hψ*(*X*) = *κ* then *e*(*X*) ^{2κ}; and if *X* is Hausdorff and *we*(*X*) = *Hψ*(*X*) = *κ*subset of a space then *e*(*X*) ^{κ}. Finally, we prove that under *V* = *L* if *X* is a first countable DCCC normal space then *X* has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in *Spaces with property* (*DC*(*ω*_{1})), *Comment. Math. Univ. Carolin.*, **58(1)** (2017), 131-135.