This is problem 5(c) in Section 24 of Munkres' Topology.
(c) Is the set $[0, 1)\times [0, 1]$ in dictionary order a linear continuum?
So, this set is basically an ordered square with the side $\{1\}\times[0, 1]$ missing. I have seen two solutions up to now, and they both claim that $C:=[0, 1)\times[0, 1]$ is a linear continuum. Recall that a linear continuum is a simply ordered set $L$ satisfying:
- $L$ has the least upper bound property
- If $x<y$, then there exists $z$ such that $x<z<y$.
I can't see why the set $C$ satisfies the first property. Take the set $S=\{(1-\frac{1}{2^n}, 1) | n\in \mathbb{N}\}$, a subset of $C$. Yet, the set does not have a least bound on dictionary order on $C$, since it would have to be $(1, 0)$ which is not on the set. Where am I missing something?