In Munkres' Topology, it's stated that if $X$ is well-ordered set then $X \times [0,1)$ is a linear continuum in the dictionary order.
Let us suppose that, $X=\{1,2\}$ and that $\mathord{<} =\{(1,2)\}$. It's clear $X$ is a well-ordered set with this order.
Now, I think that $K=\{1,2\}\times [0,1)$ is not densely ordered with the dictionary order. For $x=(1,0)$ and $y=(2,0)$ are in $K$ but there is no $z=(a,b)$ such that $x<z<y$ in the dictionary order. So, $K$ cannot be a linear continuum, contradicting what is stated in the book.
Am I wrong? If yes, where did my line of reasoning go wrong?
Your argument is incorrect, since for every $x\in[0,1)$ it holds that $(1,x)<(2,0)$.