If $X$ is a strongly zero-dimensional topological space, then $X$ is a zero-dimensional space? without the assumption $X$ completely regular

Question

I'm thinking to assum $X$ completely regular is necesary, but can't find one counterexample. A space is strongly zero-dimensional if for every pair of disjoint zero-sets of the space, there is a clopen set containing one zero setand missing the other.

Answer 1

There are many spaces $X$ such that $X$ is $T_3$ (i.e., regular and $T_1$), and every continuous real-valued function on $X$ is constant; see, for instance, this answer and Alex M.’s comment below it. Let $X$ be such a space. Then $X$ is trivially strongly zero-dimensional, since its only zero-sets are $\varnothing$ and $X$. However, it cannot be zero-dimensional: it cannot even contain a non-empty clopen set $H$ with non-empty complement, since the indicator function of $H$ would then be a non-constant continuous real-valued function on $X$.