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).