In mathematics, a Rickart space (after

Charles Earl Rickart Charles Earl Rickart (1913 – 17 April 2002) was an American mathematician, known for Rickart spaces. Rickart was born in Osage City, Kansas, and earned his B.A. and M.A. from the University of Kansas. In 1941 he received his PhD from the ...

), also called a basically disconnected space, is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

in which open σ-compact subsets have

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

open closures. named them after , who showed that Rickart spaces are related to monotone σ-complete

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...

s in the same way that

Stonean space In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is so ...

s are related to

AW*-algebra In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras, among all C*-algebras, are typically handled using one of two means: they a ...

s. Rickart spaces are

totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...

and

sub-Stonean space In topology, a sub-Stonean space is a locally compact Hausdorff space such that any two open σ-compact disjoint subsets have disjoint compact closures. Related, an F-space, introduced by , is a completely regular Hausdorff space for which every ...

References

* * Properties of topological spaces {{topology-stub