Asymmetric tie-points and almost clopen subsets of ℕ *

Alan S. Dow; Saharon Shelah

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 4, page 451-466
  • ISSN: 0010-2628

Abstract

top
A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of ℕ * . One especially important application, due to Veličković, was to the existence of nontrivial involutions on ℕ * . A tie-point of ℕ * has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of ℕ * in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.

How to cite

top

Dow, Alan S., and Shelah, Saharon. "Asymmetric tie-points and almost clopen subsets of $\mathbb {N}^*$." Commentationes Mathematicae Universitatis Carolinae 59.4 (2018): 451-466. <http://eudml.org/doc/294180>.

@article{Dow2018,
abstract = {A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of $\mathbb \{N\}^*$. One especially important application, due to Veličković, was to the existence of nontrivial involutions on $\mathbb \{N\}^*$. A tie-point of $\mathbb \{N\}^*$ has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of $\mathbb \{N\}^*$ in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.},
author = {Dow, Alan S., Shelah, Saharon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ultrafilter; cardinal invariants of continuum},
language = {eng},
number = {4},
pages = {451-466},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Asymmetric tie-points and almost clopen subsets of $\mathbb \{N\}^*$},
url = {http://eudml.org/doc/294180},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Dow, Alan S.
AU - Shelah, Saharon
TI - Asymmetric tie-points and almost clopen subsets of $\mathbb {N}^*$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 4
SP - 451
EP - 466
AB - A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of $\mathbb {N}^*$. One especially important application, due to Veličković, was to the existence of nontrivial involutions on $\mathbb {N}^*$. A tie-point of $\mathbb {N}^*$ has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of $\mathbb {N}^*$ in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.
LA - eng
KW - ultrafilter; cardinal invariants of continuum
UR - http://eudml.org/doc/294180
ER -

References

top
  1. Baumgartner J. E., Applications of the proper forcing axiom, Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 913–959. MR0776640
  2. Blass A., Shelah S., 10.1016/0168-0072(87)90082-0, Ann. Pure Appl. Logic 33 (1987), no. 3, 213–243. MR0879489DOI10.1016/0168-0072(87)90082-0
  3. van Douwen E. K., Kunen K., van Mill J., There can be C * -embedded dense proper subspaces in β ω - ω , Proc. Amer. Math. Soc. 105 (1989), no. 2, 462–470. MR0977925
  4. Dow A., Shelah S., 10.1016/j.topol.2008.05.002, Topology Appl. 155 (2008), no. 15, 1661–1671. MR2437015DOI10.1016/j.topol.2008.05.002
  5. Dow A., Shelah S., 10.4064/fm203-3-1, Fund. Math. 203 (2009), no. 3, 191–210. MR2506596DOI10.4064/fm203-3-1
  6. Dow A., Shelah S., An Efimov space from Martin's axiom, Houston J. Math. 39 (2013), no. 4, 1423–1435. MR3164725
  7. Drewnowski L., Roberts J. W., On the primariness of the Banach space l ∞ / C 0 , Proc. Amer. Math. Soc. 112 (1991), no. 4, 949–957. MR1004417
  8. Farah I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, 177 pages. Zbl0966.03045MR1711328
  9. Fine N. J., Gillman L., 10.1090/S0002-9904-1960-10460-0, Bull. Amer. Math. Soc. 66 (1960), 376–381. MR0123291DOI10.1090/S0002-9904-1960-10460-0
  10. Frankiewicz R., Zbierski P., 10.4064/fm-129-3-173-180, Fund. Math. 129 (1988), no. 3, 173–180. MR0962539DOI10.4064/fm-129-3-173-180
  11. Goldstern M., Shelah S., 10.1016/0168-0072(90)90063-8, Ann. Pure Appl. Logic 49 (1990), no. 2, 121–142. MR1077075DOI10.1016/0168-0072(90)90063-8
  12. Juhász I., Koszmider P., Soukup L., 10.1016/j.topol.2009.04.004, Topology Appl. 156 (2009), no. 10, 1863–1879. MR2519221DOI10.1016/j.topol.2009.04.004
  13. Just W., Nowhere dense P -subsets of ω , Proc. Amer. Math. Soc. 106 (1989), no. 4, 1145–1146. MR0976360
  14. Katětov M., A theorem on mappings, Comment. Math. Univ. Carolinae 8 (1967), 431–433. MR0229228
  15. Koppelberg S., 10.1007/BF00353658, Order 5 (1989), no. 4, 393–406. MR1010388DOI10.1007/BF00353658
  16. Koszmider P., Forcing minimal extensions of Boolean algebras, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117. Zbl0922.03071MR1467471
  17. Kunen K., Set Theory. An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing, Amsterdam, 1980. Zbl0534.03026MR0597342
  18. Kunen K., Vaughan J. E., eds., Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984. MR0776619
  19. Leonard I. E., Whitfield J. H. M., 10.1216/RMJ-1983-13-3-531, Rocky Mountain J. Math. 13 (1983), no. 3, 531–539. MR0715776DOI10.1216/RMJ-1983-13-3-531
  20. Pearl E., ed., Open Problems in Topology. II, Elsevier, Amsterdam, 2007. MR2367385
  21. Rabus M., On strongly discrete subsets of ω * , Proc. Amer. Math. Soc. 118 (1993), no. 4, 1291–1300. MR1181172
  22. Rabus M., An ω 2 -minimal Boolean algebra, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3235–3244. MR1357881
  23. Šapirovskiĭ B. È., The imbedding of extremally disconnected spaces in bicompacta. b -points and weight of pointwise normal spaces, Dokl. Akad. Nauk SSSR 223 (1975), no. 5, 1083–1086 (Russian). MR0394609
  24. Shelah S., Steprāns J., 10.1090/S0002-9939-1988-0935111-X, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1220–1225. MR0935111DOI10.1090/S0002-9939-1988-0935111-X
  25. Shelah S., Steprāns J., 10.1112/jlms/49.3.569, J. London Math. Soc. (2) 49 (1994), no. 3, 569–580. MR1271551DOI10.1112/jlms/49.3.569
  26. Steprāns J., 10.1090/S0002-9947-03-03329-4, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4223–4240. MR1990584DOI10.1090/S0002-9947-03-03329-4
  27. Veličković B., 10.1016/0166-8641(93)90127-Y, Topology Appl. 49 (1993), no. 1, 1–13. MR1202874DOI10.1016/0166-8641(93)90127-Y

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.