Euler characteristics – a topological invariant
One of the most unexpected developments in the twentieth century mathematics has been the meteoric rise of the subject known as topology. Topology is sometimes described as ‘rubber sheet’ geometry – a somewhat misleading description that nevertheless succeeds in capturing the flavour of the subject. Topology is the study of those properties of geometrical objects, which remain unchanged under continuous transformation of the objects. A continuous transformation is one in which points ‘close together’ to start with are ‘close together’ at the end of the cycle of transformations such as bending or stretching. What sorts of properties are topological? Certainly not the normal ones studied in Euclid’s geometry. Straightness is not a topological property. Neither is the property of being triangular. A triangle can continuously be deformed to a circle. So in topology circles and triangles are not distinguished. The basic objects studied in topology are called topological spaces. A topological space is a set endowed with some extra structure, a collection of open sets, called topology which allows us to set up idea of continuity of functions defined on the set ( look at any book of topology for a precise definition).