Why is the euler characteristic of a sphere 2?

For any triangulation of the sphere, it is true that $V-E+F=2$, where $V$ is the number of vertices in the triangulation, $E$ the number of edges in the triangulation and $F$ the number of faces in the triangulation. For example, consider the triangulation below:

There are $6$ vertices, $12$ edges and $8$ faces, so $V-E+F=6-12+8=2$.

There are also more complicated definitions of the Euler Characteristic in terms of homology or number of cells in each dimension in a CW complex. It can be defined as $$\chi(X)=\sum(-1)^n\mathrm{rank}(H_n(X))\,.$$

Consider a sequence of simple configurations on the surface of a sphere. Start with two points on the surface and connect them by two edges. This divides the surface of the sphere into two faces. Thus $\,2-2+2=2.\,$ Now remove one of the edges. This leaves only one face on the sphere. Thus $\,2-1+1=2\,$. Now move the two points together until they merge into only a single point and there is no edge. Thus $\,1-0+1=2\,$ again.