If the space $X$ is completely regular, we Know that The collection {$intZ(f)$:$f$ is a continuous function from $X$ to the real numbers} is an open base for open subsets of the space $X$ (i.e. If for each element $x$ and each open set $U_x$ of $X$, there exist a continuous real valued function $f:X→\mathbb{R}$ such that $x∈intZ(f)⊆Z(f)⊆U_x)$. I have two questions about converse of this theorem. these questions are almost the same, but I think these are different.

1.If for each element $x$ and each open set $U_x$ of $X$, there exist a continuous real valued function $f:X→\mathbb{R}$ such that $x∈intZ(f)⊆U_x$, then $X$ is completely regular.

2.If for each element $x$ and each open set $U_x$ of $X$, there exist a continuous real valued function $f:X→\mathbb{R}$ such that $x∈intZ(f)⊆Z(f)⊆U_x$, then $X$ is completely regular.

I think these two claims have counterexamples and these conditions don't emply the complete regularity of $X$.