CONTENTS 1. Interesting mappings on finite powers..................................................... 5 2. Results.................................................................................................................... 6 3. Conventions and notation.................................................................................... 8 4. βω-spaces.............................................................................................................. 8 5. Canonical partition relations and the Prime Mapping Lemma.................... 9 6. The Number of Factors Lemma......................................................................... 11 7. Consequences of the Number of Factors Lemma......................................... 13 8. Direction of the coordinate axes......................................................................... 16 9. Binary operations................................................................................................... 18 10. Extension of binary operations.......................................................................... 21 11. Stronger versions of the Prime Mapping Lemma.......................................... 22 12. Extensions of binary operations on ω............................................................... 23 13. Examples................................................................................................................ 24 14. Appendix 1: An application of non-Q-points..................................................... 25 15. Appendix 3: Homeomorphs of βω in certain finite powers............................ 26 16. Appendix 3: Mappings onto βω-spaces............................................................ 28 17. Appendix 4: Square compactiiications.............................................................. 29 18. Appendix 5: Some points of interest.................................................................. 30 References.................................................................................................................... 34
CONTENTS 0. Conventions................................................................................................................................... 5 1. Introduction............................................................................................................................................. 5 CHAPTER I. TOOLS AND THE CONSTRUCTION OF SPECIAL POINTS............................... 10 2. Tools........................................................................................................................................................ 10 3. Extension of open sets......................................................................................................................... 10 4. Construction of special points in X*................................................................................................... 12 CHAPTER II. EFFECTIVE PROOFS USING REMOTE POINTS................................................. 16 5. Remote points and extremal disconnectedness at points........................................................... 16 6. Remote points and nonhomogeneity................................................................................................ 17 7.Čech-Stone compactifications and products.................................................................................... 19 8. When "extremally disconnected at" implies remote....................................................................... 22 9. Far points, remote points, and nonhomogeneity............................................................................ 24 CHAPTER III. OTHER APPLICATIONS OF REMOTE POINTS................................................... 25 10. Two examples on extremal disconnectedness............................................................................ 25 11. A partition of R*.................................................................................................................................... 26 12. Extremal disconnectedness and C*-embedding......................................................................... 27 13. Connected compactifications........................................................................................................... 28 CHAPTER IV. PROPERTIES OF REMOTE POINTS.................................................................... 30 14: The structure of remote points.......................................................................................................... 30 15. Nonhomogeneity of spaces of special points............................................................................... 31 16. Special points and Baire spaces..................................................................................................... 32 17. Pseudocompact spaces between X and βX.................................................................................. 32 CHAPTER V. MISCELLANEOUS REMARKS................................................................................ 36 18. Zero-dimensional spaces................................................................................................................. 36 19. Q, Q*, Q**, Q***,................................................................................................................................... 37 20. Retractions from βX onto X*.............................................................................................................. 38 21. Products of remainders..................................................................................................................... 39 22. Questions............................................................................................................................................. 40 REFERENCES................................................................................................................................... 42 INDEX................................................................................................................................................... 44
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