$$A:= (-1,0) \cup\left\{ \frac {1}{n} \,\Big\vert\, n \in \mathbb N \right\} \cup\left\{ 2- {\frac {1}{n} \,\Big\vert\, n \in \mathbb N}\right\} \cup (2,3)$$
$$B:= (-1,0) \cup \left\{ 2- \frac {1}{n} \,\Big\vert\, n \in \mathbb N\right\} \cup (2,3)$$
with $\leq$ order of $\mathbb R$
Is $A$ isomorphic to $B$ ?
Cant find an isomorphism.
Any help please ?
Hint
Those sets are not order isomorphic.
By contradiction, suppose that $\phi : A \to B$ is an order isomorphism, and consider $\phi(1)$.
Derive a contradiction for each possible case $\phi(1) \in (-1,0)$, $\phi(1) \in \{ 2- \frac {1}{n} \mid n \in \mathbb N\}$ and $\phi(1) \in (2,3)$ by considering $\phi\left(\frac{1}{2}\right)$.