Definition: An ordered set $X$ is a linear continuum if the following 2 conditions hold:

• $X$ has the Least upper bound property
• $\forall x,y \in X$, if $x < y$, there exists $z \in X$ such that $x < z < y$.

Example: the Real numbers, the Ordered square

Theorem: If $X$ is a linear continiuum with the Order topology (induced by its order), then $X$ is connected, and also every interval and ray is connected. Proof