I came across the following definition of a cross section and local section in Wikipedia -
Let $\pi : E\rightarrow B$ be a fibre bundle. A cross section of this bundle is defined as a continuous map $\sigma : B\rightarrow E$ such that $\pi\circ\sigma=Id_B$ . A local section of the fiber bundle is a continuous map $\sigma : U \rightarrow E$ where $U$ is an open set in $B$ and $\pi\circ\sigma(x) = x$ for all $x \in U$ .
I have the following question -
Is a local cross section defined only using an open set or can it be defined using any subset of $B$ ? Is using an open set to define it advantageous in some way?
Thank you.