I defined a topology on $\beta{X}$, the space of all z-ultrafilters on $X$, like generated by sets $\left\{p\in\beta{X}\text{ } : A\notin{p}\right\}$ with $A$ any zero-set, and a map $i:X\rightarrow\beta{X}$ by $$i(x)=\left\{A\in\mathcal{Z}(X)\text{ }: A\in{p}\right\}.$$Where $\mathcal{Z}(X)$ is the set of all zero-sets in $X$. I proved that this map is well defined, continuous (trivial) and that if $X$ is Hausdorff and completely regular than $i$ is injective, but i should prove that $i$ is an homeomorphism between $X$ and it's image, and so for example that is open. I think it's not so hard but i can't see how to do it.
Thanks in advance.
It suffices to show that $i[C]$ is open in $i[X]$ whenever $C$ is a cozero set of $X$ (the complement of a zero set) as these form a base for the topology of $X$ and $i$ preserves arbitrary unions. To get injective let $x\neq y$ and consider a continuous function with $f(x)=0$ and $f(y)=1$.