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 Gillman & Henriksen (1956), 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 $f$ can be written as $f=g|f|$ for some real-valued continuous function $g$. 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

• Gillman, Leonard; Henriksen, Melvin (1956), "Rings of continuous functions in which every finitely generated ideal is principal", Transactions of the American Mathematical Society, 82 (2): 366–391, doi:10.2307/1993054, ISSN 0002-9947, JSTOR 1993054, MR 0078980
• 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
• Henriksen, Melvin; Woods, R. G. (1989), "F-Spaces and Substonean Spaces: General Topology as a Tool in Functional Analysis", Annals of the New York Academy of Sciences, 552 (1 Papers on General topology and related category theory and topological algebra): 60–68, Bibcode:1989NYASA.552...60H, doi:10.1111/j.1749-6632.1989.tb22386.x, ISSN 1749-6632, MR 1020774