The star countable spread and star ccc property
Volume 158 / 2019
Abstract
For a topological property $P$, say that a space $X$ is star $P$ if for every open cover $\mathcal {U}$ of $X$ there exists a subspace $A\subset X$ with the property $P$ such that $\operatorname{st} ( A,\mathcal {U} ) =X$. We analyze star ccc spaces and spaces of star countable spread describing the relationship between the respective classes as well as their place among the classes of star countable spaces, star Lindelöf spaces and feebly Lindelöf spaces. We show that, in some nice classes of spaces star countable spread and star ccc propery are equivalent to the Lindelöf property or separability. The following statements are the main results of this paper:
(i) a space $X$ is of star countable spread if and only if $X$ is star hereditarily Lindelöf;
(ii) under $\mathsf {CH}$, there is a space of star countable spread which is not star countable;
(iii) the star ccc property is not inherited by regular closed subsets even in the class of Tikhonov star countable spaces;
(iv) under $2^\omega = 2^{\omega _1}$, there exists a normal space of star countable extent which is not of star countable spread.
