An \(\alpha\)-disconnected space has no proper monic preimage. (English) Zbl 0734.54005
The authors call a continuous map \(f:X\to Y\) of compact Hausdorff spaces an \(\alpha\)-SpFi morphism (with \(\alpha\) an uncountable cardinal, or \(\alpha =\infty)\) if \(f^{-1}(G)\) is dense in X for every dense \(\alpha\)-cozero set in Y. Among the compact Hausdorff spaces, \(\alpha\)- disconnected spaces X are characterized, as follows: every monomorphism into X of the category \(\alpha\)-SpFi is one-to-one. Furthermore, a complete characterization of \(\infty\)-SpFi monomorphism is given by the authors; their result was independently obtained by R. G. Woods [Proceedings CCNY Conference on Limits 1987, Annals of New York Academy of Sciences]. Applications to Boolean algebras are also provided.
Reviewer: W.Tholen (Downsview)
MSC:
| 54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |
| 18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |
| 18B30 | Categories of topological spaces and continuous mappings (MSC2010) |
| 54C05 | Continuous maps |
| 06E15 | Stone spaces (Boolean spaces) and related structures |
| 54G10 | \(P\)-spaces |
Keywords:
extremally disconnected space; category of spaces with filters; \(\alpha\)- disconnected spacesReferences:
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