Spaces with star countable extent

For a topological property $P$, we say that a space $X$ is star $P$ if for every open cover $\mathcal{U}$ of the space $X$ there exists $A\subset X$ such that $st (A,\mathcal{U})= X$. We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf space with a point countable base which is not star countable.