Show that the set $((-1,1),\le)$ is not isomorphic with the set $((-1,0)\cup(0,1),\le)$ where $\le$ is the usual order on $\Bbb R$

Question

Show that the set $((-1,1),\le)$ is not isomorphic.with the set $((-1,0)\cup(0,1),\le)$ where $\le$ is the usual order on $\Bbb R$, $(-1,1),(-1,0),(0,1)$ are half-intervals.

My idea is that we loose all the couples $(a,0)$ and $(0,a), \forall a \in (-1,0)\cup (0,1)$ from the first set, is this proof enough? or there is stronger more formel one?

Thank you.

Answer 1

The first set has the least upper bound property.
The second set does not.