Find an element which is not in the topology.

Question

The order topology is defined on $\{0,1\} \times \mathbb{N}$. Need to find a non-singleton set which is not an element of order topology.

I Choose $A=\{(0,10), (1,1)\}$. How to prove this? Is my choice correct? We need to prove, it cannot be written as union or intersection of basis vectors?

Answer 1

$A$ is the union of two open intervals $\langle (0,9),(0,11)\rangle$ and $\langle (1,0),(1,2)\rangle$, so is in the order topology.

Observe that any open set in the order topology that contains $(1,0)$ must be infinite though. That should help.