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