Definition:Multifunction/Branch

Definition

Let $A$ and $B$ be sets.

Let $f: A \to B$ be a multifunction on $A$.

Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ such that:

$\forall i \in I: f \restriction_{A \times S_i}$ is a mapping.

Then each $f \restriction_{A \times S_i}$ is a branch of $f$.

Let $A$ and $B$ be sets.

Let $f: A \to B$ be a multifunction on $A$.

Let $\sequence {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ into branches.

It is usual to distinguish one such branch of $f$ from the others, and label it the principal branch of $f$.

Let $x \in A$ be an element of the domain of $f$.

The principal value of $x$ is the element $y$ of the principal branch of $f$ such that $\map f x = y$.

A branch point of $f$ is a point $a$ in $U$ such that:

$f$ has more than one value at one or more points in every neighborhood of $a$

$f$ has exactly one value at $a$ itself.