Definition:Section (Topology)/Mapping/Local

Definition

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

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 see

• Results about sections of mappings can be found here.

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.) ... (previous): Chapter $4$: Submersions, Immersions, and Embeddings: $\S$: Submersions