This is a problem I'm confused with:
Definition:
We say two subsets $A$ and $B$ of a topological space $X$ are completely separated if there exists a continuous map $f: X \rightarrow \mathbb{R}$ such that $f(A) \subseteq \{0\}$ and $f(B) \subseteq \{1\}$.
i.e we are not given that $A=f^{-1}(\{0\})$ but just that $f(A) \subseteq \{0\}$.
Prove that every pair of completely separated sets in a completely regular space have disjoint closures in their Stone–Čech compactification.
Right so we need to find a bounded continuous map $f: X \rightarrow \mathbb{R}$ and then extend this to $\beta(X)$. The problem is that how can we bound $f$ ? all we know is that there is a continuous map $f: X \rightarrow \mathbb{R}$ such that $f(x)=0$ if $x \in A$ and 1 if $x \in B$. But what if $f$ is unbounded in the points outside of $A$ and $B$. We are not given that $X = A \cup B$ so how can we bound $f$ ? can we simply define a new function say $g: X \rightarrow \mathbb{R}$ by $g(x) = min \{1,f(x)\}$ or something like that?
Thanks in advance.