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.
 A: No non-empty proper closed subset of $\Bbb R$ is a cozero set: a cozero set is necessarily open, since it is the inverse image of an open set under a continuous map. Every closed subset of $\Bbb R$ is a zero set, however. Similarly, every open subset of $\Bbb R$ is a cozero set, and none (except $\varnothing$ and $\Bbb R$) is a zero set.
A: This is more to add a small bit to Brian's answer.
In normal spaces, the zero sets correspond exactly to the closed G$_\delta$ subsets (and so the co-zero sets correspond exactly to the open F$_\sigma$ subsets).
The real line is perfectly normal which means in addition to normality that all closed sets are G$_\delta$ (or, equivalently, all open sets are F$_\sigma$; it is likely you have seen this second equivalent form before).  Together with the above characterisation of zero sets in normal spaces, this means that the zero sets in $\mathbb R$ correspond exactly with the closed sets (and the co-zero sets correspond exactly with the open sets); exactly as Brian has stated.
A: 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.
