In mathematics, a Rickart space (after Charles Earl Rickart), also called a basically disconnected space, is a topological space in which open σ-compact subsets have compact open closures. Grove & Pedersen (1984) named them after C. E. Rickart (1946), who showed that Rickart spaces are related to monotone σ-complete C*-algebras in the same way that Stonean spaces are related to AW*-algebras.
Rickart spaces are totally disconnected and sub-Stonean spaces.
References
Rickart space Wikipedia(Text) CC BY-SA