nLab maximal partial function

A maximal partial function is a partial function, say from $X$ to $Y$, which is maximal in the poset of partial functions from $X$ to $Y$. Explicitly, $f\colon X \nrightarrow Y$ is maximal if, given any $g\colon X \nrightarrow Y$, if $\dom f \subseteq \dom g$ and $f(a) = g(a)$ whenever $a \in \dom f$, then $\dom f = \dom g$ (and so $f = g$).

All total functions are maximal, as are all partial functions whose codomain $Y$ is the empty set. Assuming the law of excluded middle, these are the only examples. Outside of the context of constructive mathematics, therefore, (and usually in constructive math too) we are concerned with maximal partial functions subject to some restrictions, such as the maximal partial sections of some given function from $Y$ to $X$, the maximal continuous partial functions given some topological structures on $X$ and $Y$, the maximal local sections of some given continuous function from $Y$ to $X$ given some topological structures on $X$ and $Y$, etc. Then we are not working in the poset of all partial functions from $X$ to $Y$ but in a given subposet?.

More abstractly, we may also consider the maximal partial morphisms in a general category. This includes, for example, the maximal continuous partial functions between two topological spaces.