A section of a fiber bundle is a continuous map with . It chooses a point of each fibre continuously. A section of a vector bundle is the special case for vector fibres; a projective bundle section chooses a line in each fibre of the underlying vector bundle.
If the displayed fiber bundle had a section , its projection would satisfy . The cohomology ring of complex projective space is with . For a generator , write . But since , giving . This contradiction proves that no section exists, without needing the value of .
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In the context of topology and differential geometry, a **section** of a fiber bundle is a continuous function that assigns to each point in the base space exactly one point in the fiber. More formally, let's break this down: ### Fiber Bundle A **fiber bundle** consists of the following components: 1. **Base Space** \( B \): A topological space where the "fibers" are defined.