I-bundle

Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber.

The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product

, and the Möbius band is a non-trivial or twisted bundle.

Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle

Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces.

A Möbius band is a non-orientable I-bundle. The dark line is the base for a set of transversal lines that are homeomorphic to the fiber and that each touch the edge of the band twice.
An annulus is an orientable I-bundle. This example is embedded in 3-space with an even number of twists
This image represents the twisted I-bundle over the 2-torus, which is also fibered as a Möbius strip times the circle. So, this space is also a circle bundle