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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on subcontinua of $ \beta[0,\infty)-[0,\infty)$


Author: Jian-Ping Zhu
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
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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 Laver's model for the Borel conjecture.

The second answers a question posed by Baldwin and Smith negatively.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1185283-2
Keywords: Stone-Čech remainder, Laver real, continuum
Article copyright: © Copyright 1994 American Mathematical Society

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