Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in terms of volume forms.

It made me wonder if the notion of orientability can be generalized to spaces with slightly less restrictive conditions than manifolds (like the space proposed by my friend). Spaces that look locally like $\mathbb{R}^n$ in almost every point but are not a manifold (at least not with the induced topology).

Is there any useful generalization of orientability for spaces similar to manifolds but not so well behaved?

P.S. Is there a general notion of orientability, e.g. for the rationals? I saw this question but I'm precisely interested in the last point of his question, i.e. a notion generalizing the concept of orientability for manifolds for similar spaces with worse conditions.

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    $\begingroup$ For what it's worth, a Möbius strip with one segment collapsed to a point is homeomorphic to a cylinder with one generator collapsed to a point. (As for the question, I've never encountered "orientability for non-manifolds." A natural place to look would be the theory of stratified spaces. If $X$ is a topological space containing some manifold $M$ as a dense open subset, you might define $X$ to be orientable if and only if $M$ is orientable. Offhand I don't know if this is useful, however.) $\endgroup$ – Andrew D. Hwang Feb 3 '16 at 22:39
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    $\begingroup$ Maybe (I don't know) you can adapt the definition of orientability via the orientation bundle on homology manifolds. $\endgroup$ – Aloizio Macedo Feb 3 '16 at 22:48
  • $\begingroup$ If you have some sort of sensible notion of frame bundle on your space, you could define orientability via reduction of structure group. $\endgroup$ – ಠ_ಠ Feb 3 '16 at 23:40
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    $\begingroup$ @AndrewD.Hwang: I can think of situations in which it's convenient to note that singular homology can be given as a chain complex freely generated by maps out of reasonably nice stratified spaces. When writing down the differential, just like for simplices, you need to know what you're doing with orientations. So the notion is useful in this context. $\endgroup$ – user98602 Feb 4 '16 at 4:09

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