Semiseksjons

Semiseksjons is a term used in some branches of differential geometry and topology to describe a relaxation of the notion of a section of a fiber bundle. In this context, let π: E → B be a fiber bundle with base B and total space E. A semiseksjon is a map s from a subset U ⊆ B into E such that π∘s = id_U, meaning that s assigns to every point of U a point in the fiber over that point. Unlike a classical section, which is defined on all of B, a semiseksjon is defined only on U, and U may be a proper subset of B. In many treatments, U is taken to be dense in B and s is required to be continuous with respect to the subspace topology on U.

Definitions of a semiseksjon can vary slightly between authors. Some formulations include a local extension condition: for every b ∈ B and every neighborhood V of b, there exists a smaller neighborhood W ⊆ V and a map s_W defined on U ∩ W with π∘s_W = id_{U ∩ W} that agrees with s on U ∩ W. This captures the idea that a semiseksjon behaves like a section on large parts of the base while allowing gaps or singularities on a subset of B.

Examples include a global section restricted to a dense subset, or a locally defined section that cannot be extended to all of B due to topological obstructions. The concept sits between a partial section and a full section and is related to extension problems in bundle theory.

See also: section, partial section, multisection, extension problem.