As we learnt in undergraduate courses, the real line is a linear continuum, i.e. it is a totally (linearly) ordered set $X$ that satisfies the axioms:
(i) For any $x, y \in X$, if $x < y$, then there exists $z \in X$ such that $x < z < y$.
(ii) (Supremum property) Any nonempty subset $A$ of $X$ that is bounded above has a least upper bound, i.e. an element $s \in X$ such that $a < s$ for all $a \in A$ and if $x < s$ then $x$ is not an upper bound of $A$. (A least upper bound is also called a supremum.)
It can be shown that a linear continuum with more than one point has at least uncountably many points.
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