The closure of $(a;b)$ in an order topology

Question

Consider a linearly ordered set $(X; \prec)$ with its order topology.

1. Show that closure of $(a;b)$ is a subset of $[a;b]$. Under what conditions does equality hold?
2. 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.

Answer 1

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 $[2;4]$?