A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets. Also, a $T_1$ topological space $X$ such that for every closed subset $C$ of $X$ and every point $x \in X\setminus C$, there is a continuous function $f:X\rightarrow[0,1]$ such that $f(x)=0$ and $f(C)={1}$ is called completely Regular.
Now let $X$ be a completely regular totally disconnected topological space and $f$ be a real-valued continuous function over $X$( that is, $f\in C(X)$). Now if $A\subseteq X$ such that $f(A)=\{0, 1\}$, how can we define a real-valued continuous function $g$ over $X$ such that $g(X)=\{0,1\}$ and $f(a)=g(a)$ for all $a\in A$?
Best Answer
Your desired property could be reformulated as “every two subsets of $X$ that are functionally separated are clopenly separated”. This property (together with complete regularity) is called strong zero-dimensionality. In the context of Hausdorff spaces we have the following implications: \begin{align}\text{strongly zero-dimensional} \implies \text{zero-dimensional} \implies \text{totally separated} \implies \text{totally disconnected}.\end{align} The implications cannot be reversed. For example the Cantor's leaky tent with the apex removed is a metrizable totally disconnected space that is not totally separated. On the other hand, if your space $X$ is compact, then all the conditions are equivalent.