I know this is a basic question, but I have not studied homotopy theory yet. What suggested us to define a sphere with one vertex, instead of 0? How does it differ from a monotope then? This is vital in defining the Euler characteristic. Thank you in advance!
2026-03-27 23:20:11.1774653611
Why is a sphere necessarily a n-cell glued to a 0-cell?
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As part of the definition a CW-complex, the boundary of each cell must be contained in the union of all the cells of lower dimension. This means every nonempty CW-complex must have at least one $0$-cell, since otherwise there would be nowhere for the boundary of the lowest-dimensional cell to go. (This is not a problem for $0$-cells since they have no boundary.)