The closure of $(a;b)$ in an order topology Consider a linearly ordered set $(X; \prec)$ with its order topology.

*

*Show that closure of $(a;b)$ is a subset of $[a;b]$. Under what conditions does equality hold?

*Give an example of a strict inclusion.

I  solved part 1:
$$[a,b]=X \setminus((−\infty,a)∪(b,+\infty))$$
is closed and contains $(a,b)$ , so it contains the closure of $(a,b)$.
Furthermore, it equals the closure iff both endpoints are limit points of the interval.
But I cannot find example for strict inclusion.

HINT: Consider the canonical strict linear order on the natural numbers $(\mathbb{N}; <)$. 


*

*What is $(2;4)$?

*What is the closure of $(2;4)$?

*What is $[2;4]$?

