# “Agreement Domain” of Function Families

Given a family of functions $\{f_i : X \to Y\}_{i \in I}$ between topological spaces $X$ and $Y$, I define an operation $\bigcap\limits^{\scriptscriptstyle\text{dom}}$ on the family such that $\bigcap\limits^{\scriptscriptstyle\text{dom}}_{i \in I}f_i = \{x \in X\ |\ f_i(x) = f_j(x) \:\:\forall i, j \in I\}$. That is, the operation outputs the set of all points of $X$ on which every function in the family agrees on.

The notation is motivated because the operation simply takes the intersection of the graphs of each $f_i$, which is also the graph of some function $f$, then outputs the domain of $f$.

Question: Given that every $f_i$ is continuous, what restrictions must be placed on $X$, $Y$ or the family of functions itself such that $\bigcap\limits^{\scriptscriptstyle\text{dom}}_{i \in I}f_i$ is closed? For instance, perhaps compactness of $X$ would be required, or $I$ would need to be a finite index set.

A partial answer is that as long as $Y$ is Hausdorff, your "agreement domain" will be a closed subset of $X$ for any family of continuous functions $X \to Y$. (If $Y$ is Hausdorff, it is well-known that the "agreement domain" of any pair of continuous functions is closed, and the "agreement domain" for a family of continuous functions is just the intersection of all these pairwise "agreement domains".)