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SELECTION PRINCIPLES IN TOPOLOGY

This doctoral dissertation by Liljana Babinkostova delves into the evolution of selection principles in topology from the seminal works of Borel and Menger to the modern concepts like screenability and groupability. The study covers the historical background, key theorems such as Ramsey's Theorem, and new selection principles introduced by prominent mathematicians like M. Scheepers. Through a thorough examination of relations, examples, and implications, the dissertation sheds light on the relevance of selection principles in topological operations, game theory, and duality theory. The work emphasizes the importance of understanding these principles in various metric spaces and subspaces, providing a nuanced perspective on basis and measure properties. Overall, it presents a structured analysis of star-selection principles and their impact on topological research.

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SELECTION PRINCIPLES IN TOPOLOGY

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  1. SELECTION PRINCIPLES IN TOPOLOGY Doctoral dissertation by Liljana Babinkostova

  2. E. Borel 1919 Strong Measure Zero metric spaces K. Menger 1924 Sequential property of bases of metric spaces W. Hurewicz 1925 F.P. Ramsey 1930 Ramsey's Theorem F. Rothberger 1938 R.H.Bing 1951 Screenability HISTORY

  3. HISTORY • F. Galvin 1971 • R. Telgarsky 1975 • J. Pawlikovski 1994 , • Lj.Kocinac 1998 Star-selection principles • M.Scheepers 2000 Groupability

  4. Standard themes

  5. Selection principle

  6. Selection principle

  7. Selection principle

  8. C-property

  9. Screenability

  10. NEW selection principle

  11. RELATIONS

  12. RELATIONS

  13. RELATIONS Examples:

  14. Relative selection principles

  15. Relative selection principles

  16. Relative star-selection principles

  17. Equivalences and implications General Implications

  18. Equivalences and implications

  19. Equivalences and implications

  20. Equivalences and implications Star selection principles

  21. Topological operations

  22. Topological operations

  23. Game theory

  24. Assumptions Duality theory • X is a Tychonoff space • Y is a subspace of X • f is a continuous function

  25. The sequence selection property

  26. Countable fan tightness

  27. Countable strong fan tightness

  28. Strongly Frechet function

  29. (X,d) is a metric space Y is a subspace of X Basis properties Assumptions:

  30. Relative Menger basis property

  31. Relative Hurewicz basis property

  32. Relative Scheepers basis property

  33. Relative Rothberger basis property

  34. Measure properties Assumptions: (X,d) is a zerodimensional metric space Y is a subspace of X

  35. Relative Menger measure zero

  36. Relative Hurewicz measure zero

  37. Relative Scheepers measure zero

  38. http://iunona.pmf.ukim.edu.mk/~spm S P M

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