Even though this sounds plausible, it's actually false. The counterxample is the first uncountable ordinal ω₁ [1], viewed as a well-ordered set (and thus a fort

Even though this sounds plausible, it's actually false. The counterxample is the first uncountable ordinal ω₁ [1], viewed as a well-ordered set (and thus a fortiori totally ordered). For every a,b in ω₁, the closed interval [a,b] is countable (by definition), but ω₁ itself is not. [1] https://en.wikipedia.org/wiki/First_uncountable_ordinal https://en.wikipedia.org/wiki/First_uncountable_ordinal