Can an uncountable set be a "chain"?

Question

In the Wikipedia page about total orders, it's stated that a "chain" is a synonym for a totally ordered set. But this makes no sense to me, since "chain" seems to suggest countability, and yet the real numbers would be a chain because they are totally ordered by "less than or equal to." Am I missing something, or is it true that "chain" can refer to uncountable sets?

Answer 1

"Chain" can indeed refer to an uncountable set; for instance, $\mathbb{R}$ is a chain in $(\mathbb{R}, <)$.

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For instance, Zorn's Lemma (https://en.wikipedia.org/wiki/Zorn%27s_lemma) is usually stated as "If every chain in $\mathbb{P}$ has an upper bound, then $\mathbb{P}$ has a maximal element"; in most interesting cases, most chains in $\mathbb{P}$ are uncountable. For example, when using Zorn's lemma to prove that every set can be well-ordered, the partial order involved is the set of all partial well-orderings of a given set, and this is usually really uncountable.

Answer 2

Of course it can. Consider $\{(-r,r)\mid r\in \Bbb R \text{ and }r>0\}$.