I'm looking for info on why E is commonly used for the total space of a fiber bundle. I understand F (fiber) and B (base), but there doesn't seem to be any particularly obvious reason for choosing E.
Early contributors to the theory of fiber bundles, like Ehresmann who developed the geometric formalism of general connections, were French. In French total space is espace étalé. In more refined terminology espace étalé is something more structured, which only coincides with the total space as a set, and applies to sheaves, which generalize fiber bundles. This generalization was also developed by the French, espace étalé definition was used in the 1950 Cartan seminar.