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 finitely generated ideal of the ring of real-valued continuous functions is principal, or equivalently every real-valued continuous function  can be written as  for some real-valued continuous function . When dealing with compact spaces, the two concepts are the same, but in general, the concepts are different. The relationship between the sub-Stonean spaces and F-spaces is studied in Henriksen and Woods, 1989. Examples
Rickart spaces and the corona sets of locally compact σ-compact Hausdorff spaces are sub-Stonean spaces. See also
 Extremally disconnected space
 F-space

References

Properties of topological spaces