Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
ultrafilter; cardinal invariants of continuum
ultrafilter; cardinal invariants of continuum
Summary:
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.
References:
[1] Baumgartner J. E.: Applications of the proper forcing axiom. Handbook of Set-theoretic Topology, North-Holland Publishing, Amsterdam, 1984, pages 913–959. MR 0776640
[2] Blass A., Shelah S.: There may be simple $P_{\aleph_1}$- and $P_{\aleph_2}$-points and the Rudin-Keisler ordering may be downward directed. Ann. Pure Appl. Logic 33 (1987), no. 3, 213–243. DOI 10.1016/0168-0072(87)90082-0 | MR 0879489
[3] van Douwen E. K., Kunen K., van Mill J.: There can be $C^*$-embedded dense proper subspaces in $\beta\omega-\omega$. Proc. Amer. Math. Soc. 105 (1989), no. 2, 462–470. MR 0977925
[4] Dow A., Shelah S.: Tie-points and fixed-points in $\mathbb N^*$. Topology Appl. 155 (2008), no. 15, 1661–1671. DOI 10.1016/j.topol.2008.05.002 | MR 2437015
[5] Dow A., Shelah S.: More on tie-points and homeomorphism in $\mathbb N^\ast$. Fund. Math. 203 (2009), no. 3, 191–210. DOI 10.4064/fm203-3-1 | MR 2506596
[6] Dow A., Shelah S.: An Efimov space from Martin's axiom. Houston J. Math. 39 (2013), no. 4, 1423–1435. MR 3164725
[7] Drewnowski L., Roberts J. W.: On the primariness of the Banach space $l_{\infty}/C_0$. Proc. Amer. Math. Soc. 112 (1991), no. 4, 949–957. MR 1004417
[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. MR 1711328 | Zbl 0966.03045
[9] Fine N. J., Gillman L.: Extension of continuous functions in $\beta N$. Bull. Amer. Math. Soc. 66 (1960), 376–381. DOI 10.1090/S0002-9904-1960-10460-0 | MR 0123291
[10] Frankiewicz R., Zbierski P.: Strongly discrete subsets in $\omega^*$. Fund. Math. 129 (1988), no. 3, 173–180. DOI 10.4064/fm-129-3-173-180 | MR 0962539
[11] Goldstern M., Shelah S.: Ramsey ultrafilters and the reaping number---$ Con({\mathfrak r}<{\mathfrak u})$. Ann. Pure Appl. Logic 49 (1990), no. 2, 121–142. DOI 10.1016/0168-0072(90)90063-8 | MR 1077075
[12] Juhász I., Koszmider P., Soukup L.: A first countable, initially $\omega_1$-compact but non-compact space. Topology Appl. 156 (2009), no. 10, 1863–1879. DOI 10.1016/j.topol.2009.04.004 | MR 2519221
[13] Just W.: Nowhere dense $P$-subsets of $\omega$. Proc. Amer. Math. Soc. 106 (1989), no. 4, 1145–1146. MR 0976360
[14] Katětov M.: A theorem on mappings. Comment. Math. Univ. Carolinae 8 (1967), 431–433. MR 0229228
[15] Koppelberg S.: Minimally generated Boolean algebras. Order 5 (1989), no. 4, 393–406. DOI 10.1007/BF00353658 | MR 1010388
[16] Koszmider P.: Forcing minimal extensions of Boolean algebras. Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117. MR 1467471 | Zbl 0922.03071
[17] Kunen K.: Set Theory. An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing, Amsterdam, 1980. MR 0597342 | Zbl 0534.03026
[18] Kunen K., Vaughan J. E., eds.: Handbook of Set-theoretic Topology. North-Holland Publishing, Amsterdam, 1984. MR 0776619
[19] Leonard I. E., Whitfield J. H. M.: A classical Banach space: $l_{\infty }/c_{0}$. Rocky Mountain J. Math. 13 (1983), no. 3, 531–539. DOI 10.1216/RMJ-1983-13-3-531 | MR 0715776
[20] Pearl E., ed.: Open Problems in Topology. II. Elsevier, Amsterdam, 2007. MR 2367385
[21] Rabus M.: On strongly discrete subsets of $\omega^\ast$. Proc. Amer. Math. Soc. 118 (1993), no. 4, 1291–1300. MR 1181172
[22] Rabus M.: An $\omega_2$-minimal Boolean algebra. Trans. Amer. Math. Soc. 348 (1996), no. 8, 3235–3244. MR 1357881
[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). MR 0394609
[24] Shelah S., Steprāns J.: PFA implies all automorphisms are trivial. Proc. Amer. Math. Soc. 104 (1988), no. 4, 1220–1225. DOI 10.1090/S0002-9939-1988-0935111-X | MR 0935111
[25] Shelah S., Steprāns J.: Somewhere trivial autohomeomorphisms. J. London Math. Soc. (2) 49 (1994), no. 3, 569–580. DOI 10.1112/jlms/49.3.569 | MR 1271551
[26] Steprāns J.: The autohomeomorphism group of the Čech-Stone compactification of the integers. Trans. Amer. Math. Soc. 355 (2003), no. 10, 4223–4240. DOI 10.1090/S0002-9947-03-03329-4 | MR 1990584
[27] Veličković B.: $ OCA$ and automorphisms of ${\mathscr P }(\omega)/ fin$. Topology Appl. 49 (1993), no. 1, 1–13. DOI 10.1016/0166-8641(93)90127-Y | MR 1202874
Search
Partner of
