A Note on Stone Spaces

Abstract

The goal of this paper is to characterize each of compact, totally disconnected, Stone relation spaces, and Stone reflexive spaces as well as examine the relationships between them. Finally, we investigate some properties of them and compare our results.

Keywords

• Reflexive spaces
• Stone spaces
• Total disconnected spaces
• Hausdorff spaces

References

1. A.V. Arkhangel'skii, V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, Reidel (Translated from Russian), (1984).
2. M. Baran, Separation properties, Indian J. Pure Appl. Math., 23, 333-341, (1991).
3. M. Baran and H. Altindis, T_2-objects in topological categories, Acta Math. Hungar., 71, 41-48, (1996).
4. M. Baran, A Notion of compactness in topological categories, Publ. Math. Debrecen, 50, 221-234, (1997).
5. M. Baran, PreT_2 objects in topological categories, Appl. Categor. Struct., 17, 591-602, (2009).
6. M. Baran, Separation, connectedness and disconnectedness, Turk J Math, 47, 279-295, (2023).
7. M. Baran, Stone spaces I, submitted to Filomat.
8. M. M. Clementino, E. Giuli, W. Tholen, Topology in a category: Compactness, Portugal. Math., 53, 129-143, (1996).

9. D. Dikranjan, W. Tholen, Categorical Structure of Closure Operators, Dordrecht, Netherlands: Kluwer Academic Publishers, (1995).
10. R. Engelking, General topology, Heldermann Verlag, Berlin (1989).
11. H. Herrlich, G. Salicrup, G.E. Strecker, Factorizations, denseness, separation, and relatively compact objects, Topology Appl., 27, 157-169, (1987).
12. P.T. Johnstone, The Gleason cover of a topos I, J. Pure Appl. Algebra, 19, 171-192, (1980).
13. P.T. Johnstone, Stone Spaces, Cambridge Univ. Press, (1982).
14. E.G. Manes, Compact Hausdorff objects, General Topology Appl., 4, 341-360, (1974).
15. M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41, 375-481, (1937).
16. M.H. Stone, Algebraic characterization of special Boolean rings, Fund. Math., 29, 223-303, (1937).