We begin with a combinatorial object, namely a planar network that is of fundamental in building up geometric objects and discuss the notion of euler characteristics associated with it. A network consists of

1. A set N, of points in a plane, whose elements are called vertices.
2. A way of specifying when two vertices are joined together by non-intersecting lines.

Thus a network can be described by dots representing the vertices and joining a pair of vertices by non-intersecting lines called edges, when they are joined together.

A path in a network between two vertices a and b is a sequence of edges, starting at a and ending at b, such that each end ends where the next one begins.

A network is called connected if any two vertices can be joined by a path. A network is finite if the set of vertices and the set of edges are finite. In this discussion all networks are assumed to be finite. Such a number divides the plane into a number of regions of which the bounded ones are called faces of the network. For any network N, let F, E, V denote respectively the number of faces, the number of edges and the number of vertices. If you draw some networks and compute V-E+F, you will see that this is always 1. The first person to prove that this formula does in fact hold for any network was Euler. Thus if one changes a given network to another by removing certain vertices or edges, V-E+F remains unchanged.