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Given set $\mathcal{P}$ of subsets of a countable set $X$. For each $A, B \in \mathcal{P}$ it is given that $A \subset B$ or $B \subset A$. Does it follow that $\mathcal{P}$ is countable itself?

Asaf Karagila
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Mikhail Kuzin
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  • More or less a duplicate: http://math.stackexchange.com/questions/253192/chain-of-length-2-aleph-0in-p-mathbbn-subseteq – Asaf Karagila Jul 15 '15 at 14:24

1 Answers1

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No. For each $x\in\Bbb R$ let $A_x=\{q\in\Bbb Q:q<x\}$.

Brian M. Scott
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