A Note on Minimal Separating Function Sets

We study point-separating function sets that are minimal with respect to the property of being separating. We first show that for a compact space $X$ having a minimal separating function set in $C_p(X)$ is equivalent to having a minimal separating collection of functionally open sets in $X$. We also identify a nice visual property of $X^2$ that may be responsible for the existence of a minimal separating function family for $X$ in $C_p(X)$. We then discuss various questions and directions around the topic.