Any open set in the topology generated by clopen basis.

Question

Let $(X,\tau)$ be a topological space generated by a clopen basis.

Then, is it true that every element in $\tau$ is clopen?

It could be false for the complement of arbitrary union of base elements.

But, i'm not sure about that.

Give some comment. Thank you!

Answer 1

Consider $\mathbb Q$, endowed with its usual topology. That topology is generated by the sets of the form $(c,d)\cap\mathbb Q$, with $c<d$ and $c,d\in\mathbb{R}\setminus\mathbb{Q}$, which are clopen. However, not all subsets of $\mathbb Q$ are clopen.