A note on subcontinua of $\beta [0,\infty )-[0,\infty )$
HTML articles powered by AMS MathViewer
- by Jian-Ping Zhu PDF
- Proc. Amer. Math. Soc. 120 (1994), 597-602 Request permission
Abstract:
Let $M = { \oplus _{n \in \omega }}{I_n}$ be the topological sum of countably many copies of the unit interval $I$. For any ultrafilter $u \in {\omega ^{\ast }}$, we let ${M^u} = \cap \{ {\operatorname {cl} _{\beta M}}( \cup \{ {I_n}:n \in A\} ):A \in u\}$. It is well known that ${M^u}$ is a decomposable continuum with a very nice internal structure. In this paper, we show: (1) every nondegenerate subcontinuum of $\beta [0,\infty ) - [0,\infty )$ contains a copy of ${M^u}$ for some $u \in {\omega ^{\ast }}$. (2) there is no nontrivial simple point in Laverβs model for the Borel conjecture. The second answers a question posed by Baldwin and Smith negatively.References
- Stewart Baldwin and Michel Smith, On a possible property of far points of $\beta [0,\infty )$, Proceedings of the 1986 Topology Conference (Lafayette, LA, 1986), 1986, pp.Β 239β245. MR 945501
- David P. Bellamy, A non-metric indecomposable continuum, Duke Math. J. 38 (1971), 15β20. MR 271911
- Andreas Blass and Saharon Shelah, 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.Β 3, 213β243. MR 879489, DOI 10.1016/0168-0072(87)90082-0 E. K. van Douwen, Subcontinua and nonhomogeneity of $\beta {\mathbb {R}^ + } - {\mathbb {R}^ + }$, Notices Amer. Math. Soc. 24 (1977), A-559.
- Eric K. van Douwen, The number of subcontinua of the remainder of the plane, Pacific J. Math. 97 (1981), no.Β 2, 349β355. MR 641164, DOI 10.2140/pjm.1981.97.349
- Haim Judah, Saharon Shelah, and W. H. Woodin, The Borel conjecture, Ann. Pure Appl. Logic 50 (1990), no.Β 3, 255β269. MR 1086456, DOI 10.1016/0168-0072(90)90058-A
- Richard Laver, On the consistency of Borelβs conjecture, Acta Math. 137 (1976), no.Β 3-4, 151β169. MR 422027, DOI 10.1007/BF02392416
- Arnold W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), no.Β 1, 93β114. MR 613787, DOI 10.1090/S0002-9947-1981-0613787-2
- Jerzy Mioduszewski, On composants of $\beta R-R$, Proceedings of the Conference Topology and Measure, I (Zinnowitz, 1974) Ernst-Moritz-Arndt Univ., Greifswald, 1978, pp.Β 257β283. MR 540576
- Michel Smith, $\beta ([0,\infty ))$ does not contain nondegenerate hereditarily indecomposable continua, Proc. Amer. Math. Soc. 101 (1987), no.Β 2, 377β384. MR 902559, DOI 10.1090/S0002-9939-1987-0902559-8
- Michel Smith, The subcontinua of $\beta [0,\infty )-[0,\infty )$, Proceedings of the 1986 Topology Conference (Lafayette, LA, 1986), 1986, pp.Β 385β413. MR 945509
- Jian-Ping Zhu, On indecomposable subcontinua of $\beta [0,\infty )-[0,\infty )$, Proceedings of the Tsukuba Topology Symposium (Tsukuba, 1990), 1992, pp.Β 261β274. MR 1180813, DOI 10.1016/0166-8641(92)90008-N
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 597-602
- MSC: Primary 54D40; Secondary 03E35, 54F15
- DOI: https://doi.org/10.1090/S0002-9939-1994-1185283-2
- MathSciNet review: 1185283