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The projection map in a fiber bundle sends each point of the total space to the corresponding point in the base space, essentially 'collapsing' the fibers to points.

A section of a fiber bundle is a continuous map from the base space to the total space such that the composition with the projection map is the identity on the base space.

The structure group of a fiber bundle is a topological group that acts on the fiber space in such a way that the local trivializations are compatible with this action.

The total space of a fiber bundle is the space that 'contains' all fibers, each corresponding to a point in the base space. It is where the bundle's topology is primarily considered.

A fiber bundle being locally trivial means that around every point in the base space, there is a neighborhood that has the structure of a product space with the fiber, although this may not be the case globally.

A fiber bundle is a space that is locally a product space but globally may have a different topological structure. It consists of a total space, a base space, a fiber space, and a projection map.

A trivial fiber bundle is a fiber bundle where the total space is globally a product of the base space and the fiber space.

Fibers in a fiber bundle are the spaces that, when projected down by the projection map, correspond to a single point in the base space. Each fiber is homeomorphic to the fiber space and varies smoothly or continuously across the base space.

A vector bundle is a fiber bundle where the fiber is a vector space and the structure group is a linear group, allowing for vector space operations in each fiber that vary continuously across the bundle.

A local trivialization of a fiber bundle is a homeomorphism from a neighborhood in the total space to a product of a neighborhood in the base space with the fiber space, mapping fibers to fibers.

Transition functions are functions that describe how local trivializations change from one neighborhood to another in the overlap region, ensuring that the fiber bundle structure is well-defined.