A note on subcontinua of 𝛽[0,∞)-[0,∞)

Zhu, Jian-Ping

doi:10.1090/S0002-9939-1994-1185283-2

A note on subcontinua of $\beta [0,\infty )-[0,\infty )$
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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 Laver’s model for the Borel conjecture. The second answers a question posed by Baldwin and Smith negatively.

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Subcontinua and nonhomogeneity of

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