Proof Verification Half Open Intervals are Normal Topological Spaces

Question

This is the question I am attempting: Show that the real line with the topology generated by the collection of half-open intervals $[a, b) = \{x \mid a \leq x < b \}$ is normal topological space.

Here is my attempt at the question. My proof seems too simple so I am guessing that I missed something. Any help is appreciated.

Proof: Here the basis elements of this topology are half-open intervals. So any open set must be some combination of those basis elements. So any open set $U$ has the following form. $$ U = \cdots \cup [a_i, b_i) \cup [a_{i+1}, b_{i+1}) \cup \cdots $$

So, the closed set $C = X \setminus U$ would have the form $$ C = \cdots \cup [b_i, a_{i+1}) \cup [b_{i+1}, a_{i+2}) \cup \cdots $$

But notice that this is also a combination of basis elements. So $C$ is also open. This implies that any closed set is also open and vice versa. So if we have two disjoint closed sets $C_1, C_2$, then trivially $C_1, C_2$ are disjoint open neighborhoods containing $C_1, C_2$, meaning that $X$ is indeed a normal topological space.