# Unique extendable functions… Is there a theory?

I also made this post in MSE, but I think it may fit here as well.

Motivation: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function from $X$ to $Y$ that coincides with $f$ in a dense set, then $g=f$.

Take an analytic function $f: \mathbb{C} \rightarrow \mathbb{C}$. If $g$ is an analytic function that coincides with $f$ in a set with limit point, then $g=f$.

Okay, now given the motivation, my question is the following:

Is there a theory which has the objective of characterizing functions that satisfy a certain "unique extension" property, in a general context?

For example, we could define:

Definition: Given a family $\mathcal{F}$ of maps $f:X \rightarrow Y$ , $f \in \mathcal{F}$ is said to satisfy $P$-unique-extendability on $(X,Y)$ if for every subset $A \subset X$ which satisfies the property $P$, we have that every function $g \in \mathcal{F}$ which coincides with $f$ in $A$ is such that $f=g$.

Therefore, for example, every continuous function is dense-unique-extendable on $(\mathbb{R}$, $\mathbb{R})$

Maybe there is a categorical POV of this?

• The only thing that comes to mind is the categorical notion of epimorphism. It is unclear to me what you expect a general theory to do; in such generality, it seems very unlikely to me that you can actually say anything interesting. – Eric Wofsey May 31 '15 at 16:31
• Sorry if I was unclear. I agree with you, but what I mean is: maybe searching for those things in spaces we already know, in order to characterize objects which satisfy it, could be something fruitful. – Aloizio Macedo May 31 '15 at 16:55