TITLE: Characteristic classes, Möbius strip, and the cylinder
QUESTION [6 upvotes]: I have been thinking how to distinguish the (open) Möbius strip from a(n open) cylinder.
What does not work

Standard invariants from general topology, as connectedness or compactness,
Invariants that depend on the mere homotopy type of the space, as homotopy groups and (co)homology.

What does work
The only invariant I can think of is orientability. My question is therefore:

Is there any other invariant that can be used to show that the Möbius strip and a cylinder are not homeomorphic?

If we regard both spaces as line bundles over the circle, we can show with the Stiefel–Whitney classes that they are non-isomorphic vector bundles, but this seems to be a weaker statement that being non-homeomorphic. (And moreover Stiefel–Whitney can be treated as a reformulation of the orientability argument).

REPLY [7 votes]: As an alternative to James's answer, you can look at the one-point compactifications. For the cylinder, you get a space homeomorphic to a sphere with two points identified, which has $\Bbb{Z}$ for its fundamental group. For the Möbius strip you get a space homeomorphic to the real projective plane, which has $\Bbb{Z}/2\Bbb{Z}$ for its fundamental group.