graphics grabbed from Lawson 03
The Möbius strip is the topological space obtained from the “open strip”, hence the square, by gluing two of its opposite sites but after applying half of a full rotation to one of them
Regarded as a manifold, then the Möbius strip is among the simplest examples of manifolds which is not orientable. Regarding as a real vector bundle over the circle, then the Möbius strip is among the simplest examples of a non-trivial vector bundle.
As a topological space, the Möbius strip is
the quotient topological space obtained from the square $[0,1]^2$ by the equivalence relation which identifies two of the opposite sides, but with opposite orientation
For the realization of the Möbius strip as a topological vector bundle (a real line bundle) see there, this example.
Named after August Möbius.