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So proofwiki, and some other sites, claim the Euler-characteristic of the Möbius strip is 0-1+1=0. Relying on the fact that a Möbius strip has no vertices, i.e. 0-cells.

However I can nowhere find a CW-complex without 0-cells that would result in the Möbius strip. Even more so this question directly says that any meaningful CW-complex has at least 1 0-cell.

So I lack the understanding how the Möbius-band has no vertices if in the core of CW-complexes every non-empty space has at least 1 0-cell?

Is vertex=0-cell a simplification, or am I missing something else?

Lay
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    Every nonempty CW complex has at least one 0-cell, as you say. – John Palmieri Nov 08 '22 at 17:23
  • I guess my question isn't really clear to myself either.

    Perhaps I should rephrase as: Clearly there exists no CW-complex without 0-cells. So how come people can claim the Möbius band has 0 vertices?

     – Lay Nov 08 '22 at 17:36
  • Probably the best way to answer that question is to ask the people making the claim. Note also that if they're doing things the "obvious" way — the single 1-cell is the boundary of the Möbius band — then the boundary of the 2-cell will not lie in the 1-skeleton. So there are other problems with their cell structure. Note also that they don't claim that their structure is a CW-decomposition. Maybe they have some other structure in mind? – John Palmieri Nov 08 '22 at 17:48
  • I think my understanding of CW-complexes is too little, as I don't see why the boundary of the 2-cell wouldn't correspond to the circle 1-cell surrounding the 2-cell with a loop.

    I guess they indeed would be the best to ask, but I'm afraid I won't be able to ever reach them. Would you know of/hint at a structure which would be more in line with their reasoning and/or are there other ways to define a vertex other than via CW-complexes?

     – Lay Nov 08 '22 at 18:20
  • The region bounded by the circle is not a 2-cell: its interior is not homeomorphic to an open disk. Regarding their reasoning: I have no idea. I think they're just wrong. – John Palmieri Nov 08 '22 at 19:26
  • Ah of course: thanks that cleared up a lot. And thank you for your time and effort in commenting. – Lay Nov 08 '22 at 19:45
  • @Lay You should now write an answer to yourown question. – Paul Frost Nov 09 '22 at 00:00
  • @PaulFrost, ah okay thanks, forget about that. – Lay Nov 09 '22 at 11:28

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I think the conclusion is that proofwiki and other sources which claim the möbius band has 0 vertices, 1 face and 1 edge, thus calculating $\chi = V - E + F = 0-1+1=0$ for the Euler characteristic, use a more instinctual definition of what a vertex, edge and face are.

A, non-empty, CW-complex has at least 1 0-cell, which intuitively is often seen as a point/vertex, so calculating $\chi$ using CW-complexes will result in a calculation where $\sigma_{0} \geq 1$, where $\sigma_0$is now meant to be the number of 0-cells in the chosen CW-complex; which by the above does not need to correspond with the instinctive idea of a vertex.

Lay
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