I want to prove that the interval $[a,b)\subset \mathbb{R}$ is not closed using the definition that a set $A$ in a topological space $X$ is closed iff its complement $X-A$ is open.
Here, the topology on $R$ is the standard order topology.
Munkres in his book "Topology" mentions the claim in passing right after he defines what it means for a set to be closed.
My attempts was:
Assume to the contrary that $[a,b)$ is closed. Then, by definition, its complement $(-\infty,a)\cup[b,+\infty)$ is open. I know that $(-\infty,a)$ is open while $[b,+\infty)$ is closed, but it does not give me a contradiction since the union of an open and a closed need not be not open.
How should I proceed?
I want a topological proof that does not involve epsilon balls and even sequences since the book has not discussed sequences yet.