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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.

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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".)

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  • $\begingroup$ Indeed, by searching on Hausdorff Spaces I was able to find out about Equalizers which are the operations which find "agreement domains". $\endgroup$ – Herng Yi Oct 5 '12 at 14:32
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I remark that in the language of category theory, the "agreement domain" could be called the equalizer of the family. See here for the standard definition, which only needs to be modified a bit to yield the notion you're interested in.

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