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?
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.