Spaces with a pretty base (Q1174822)

Let \(R\) be a reduced commutative ring satisfying the countable annihilator condition. By earlier results of the author with \textit{M. Jerison} [Trans. Am. Math. Soc. 115, 110--130 (1965; Zbl 0147.29105)] the space of minimal prime ideals of \(R\) (with the hull-kernel topology) has a ``pretty base'', i.e. a base of clopen sets that is closed under the formation of closures of countable unions. In this paper topological spaces with pretty bases are studied in the abstract. In particular, their connection to basically disconnected spaces, and the properties of their weakly Lindelöf subspaces, are investigated.