Zero and Cozero-sets of $\mathbb{R}$ A subset $U$ of a space $X$ is said to be a zero-set if there exists a continuous real-valued function $f$ on $X$ such that $U=\{x\in X: f(x)=0\}$. and said to be a Cozero-set if here exists a continuous real-valued function $g$ on $X$ such that $U=\{x\in X: g(x)\not=0\}$.
Is it true that every closed set in $\mathbb{R}$ is a Cozero-set?
I guess since $\mathbb{R}$ is a completely regular this implies that every closed set is Cozero-set, but by the same argument use completely regular property on $\mathbb{R}$, every closed subset of $\mathbb{R}$ is a zero-set. This argument is correct?
How can we discussed the relation between open & closed subset of $\mathbb{R}$ and zero and cozero-sets?
thanks.

I just waant to know how $\phi$ and $\mathbb{R}$ are not  zero set?
as if i take $f(x) =  0  \forall x$ and $g(x)  =  e^{x} + 1 \forall x$
both are cts.
Then $\phi$ and $\mathbb{R}$ are zero set.
