Definition for Euler characteristic without CW-complexes It is possible to have a definition of the Euler characteristic without using CW-complexes? (I'm referring to the definition given by Wikipedia :  http://en.wikipedia.org/wiki/Euler_characteristic#Topological_definition.)
What is the geometrical picture behind these numbers, of there is any?
 A: Yes, as we can take singular homology (which we can for all topological spaces), then we define the Betti numbers to be the ranks of the individual homology groups and then if the alternating sum of Betti numbers of the space is finite, then we define the Euler characteristic to be this alternating sum. The Euler characteristic is undefined if this alternating sum does not converge - this definition can be extended to include other spaces but at the cost of properties of the Euler characteristic.
With regard to your geometric picture question, the Euler characteristic is mostly a book-keeping tool, although it certainly is related to geometric properties, mostly through the genus of a surface, but also in how we can manipulate the Euler characteristic through, for example, disjoint union which corresponds to addition of Euler characteristic. We also have that the wedge product corresponds to addition minus 1, the product of spaces corresponds to the product of Euler characteristics, and more generally this holds for a fibration where the Euler characteristic of the total space is the product of the Euler characteristics of the base space and fiber.
