Rickart space

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 under Gelfand duality, in the same way that Stonean spaces are related to AW*-algebras.

Rickart spaces were also studied by Paul Halmos under the name Boolean σ-spaces, as they correspond to Boolean σ-algebras via Stone duality. The concept of Rickart spaces resurfaced in Jamneshan & Tao (2023) under the name Stone_σ-spaces.

Both algebraic descriptions (namely, the C*-algebraic and Boolean algebraic ones) are explicitly discussed in Fritz & Lorenzin (2025).

Rickart spaces are totally disconnected and sub-Stonean spaces.

References

• Grove, Karsten; Pedersen, Gert Kjærgård (1984), "Sub-Stonean spaces and corona sets", Journal of Functional Analysis, 56 (1): 124–143, doi:10.1016/0022-1236(84)90028-4, ISSN 0022-1236, MR 0735707
• Rickart, C. E. (1946), "Banach algebras with an adjoint operation", Annals of Mathematics, Second Series, 47 (3): 528–550, doi:10.2307/1969091, ISSN 0003-486X, JSTOR 1969091, MR 0017474
• Halmos, Paul (1974), Lectures on Boolean Algebras, Undergraduate Texts in Mathematics, Springer-Verlag New York Inc., doi:10.1007/978-1-4612-9855-7, ISBN 978-1-4612-9855-7
• Jamneshan, Asgar; Tao, Terence (2023), "Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration", Fundamenta Mathematicae, 261: 1–98, doi:10.4064/fm226-7-2022
• Fritz, Tobias; Lorenzin, Antonio (2025), "Categories of abstract and noncommutative measurable spaces", arXiv:2504.13708 [math.OA]