Definition:Section (Topology)

Definition

Let $S$ and $T$ be topological spaces.

Section of Mapping

Let $\phi: S \to T$ be a continuous surjective mapping.

Let $I_T: T \to T$ be the identity mapping on $T$.

Then a section of the mapping $\phi$ is a mapping $\sigma: T \to S$ which is a continuous right inverse of $\phi$:

$\sigma: T \to S \text { such that } \phi \circ \sigma = I_T$

Local Section of Mapping

Let $U \subseteq T$ be an open set of $T$.

Let $\phi: S \to T$ be a continuous surjective mapping.

Let $I_U: U \to U$ be the identity mapping on $U$.

Then a local section of the mapping $\phi$ is a mapping $\sigma: U \to S$ which is a continuous right inverse of $\phi$:

$\sigma: U \to S \text { such that } \phi \circ \sigma = I_U$

Also known as

Some authors use the word cross section rather than section.

Also see

• Definition:Fiber Bundle

• Results about sections in the context of topology can be found here.

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): $\S 10$: Local and Global Sections of Vector Bundles