My professor said that this (picture) is an example for a non-cell-decomposition because there is missing an vertex at the end of the edge. He also claimed that we have $2$-faces, $3$-edges and $2$-vertices.
I can't understand both arguments. Also I only count $1$-face, $2$-edges and $1$-vertex. Can someone explain the two arguments?
Thank you in advance.
It's a bit difficult talking about this stuff without drawing tools.
You counted $2$ edges. I guess they are the circle and the vertical line.
For the circle, the two end-points coincide at the (bold) vertex. Where are the end-points of the vertical line? Note that end-points of $1$-cells must be $0$-cells in a cell decomposition.