TITLE: Uncountable chain of subsets of $\mathbb N$
QUESTION [1 upvotes]: Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for  every pair of elements in $E$, one is a subset of the other. Clearly, $E$ could be countable by considering $E=\big\{\{1,\cdots, n\}, n\in\mathbb N\big\}$. My question is the following: could we find an example that $E$ is uncountable? Many thanks for the answer!

REPLY [6 votes]: This is indeed possible. Firstly note that if we replace $\Bbb N$ with $\Bbb Q$ we will get an equivalent problem, because there exists a bijection between $\Bbb N$ and $\Bbb Q$. Now we can take $E=\{S_r:r\in\Bbb R\}$, where $S_r=\{q\in\Bbb Q:q<r\}$. You should be able to see yourself why this works.
{"set_name": "stack_exchange", "score": 1, "question_id": 1266426}