euler characteristic

algebraic topology attaches algebraic structures and invariants of these to spaces. this is usually done by dividing the space into chambers which are themselves simple like rooms.one has besides the chambers, the walls , the corner lines and the corner points. One calls these respectively in reverse order. 0 cells. 1 cells 2 cells 3 cells etc. the goal is to find structures that are essentially unchanged when a space divided into cells is subdivided into smaller cells.

the earliest and most famous such invariant is the alternating sum of the number of k cells:

#of zero cells - #of 1 cells + #of 2 cells ... which sum terminates for a space divided into finitely many cells.

there is exactly one more independent topological invariant of this simple form. namely it only depends on the number of cells of each type and not how they hook together...

problem: notice it and prove this claim.