Does $T_3$ imply that the topological space is zero-dimensional?

Question

First, let me clear up my definitions:

1. A topological space is $T_3$ if, given any point $x$ and closed set $F$ in $X$ such that $x$ does not belong to $F$, they are separated by neighbourhoods.
2. A topological space is zero-dimensional if it has a basis from clopen sets.

Now, it is not hard to show that if a topological space is zero-dimensional, it is also $T_3$. However, I am wondering if the converse is also true; does every $T_3$ space have a basis from clopen sets? If that is the case, how can I prove it and if it isn't, can you provide me a counter-example and perhaps a slightly modified implication (perhaps with some additional requirement) that would hold?

Answer 1

No; $\mathbb{R}$ is a counterexample because its only clopen sets are $\emptyset$ and $\mathbb{R}$ itself. In fact, any connected metric space provides a counterexample.