A topological transformation or a homeomorphism f: X – Y between two spaces X and Y is a bijection such that both f and f-inv are continuos. If such a function exists, X and Y are said to be topologically equivalent. A topological property is a property of a space X which is shared by any space homeomorphic to X. topological properties are called topological invariants. The aim of the article is to describe an age-old topological invariant – Euler characteristics, which remains an important notion with the development of other invariants. It is usually not too hard to prove that two given topological spaces are topologically equivalent (assuming this to be the case). All we require is to exhibit a homeomorphism between the spaces. Much harder is the proof that two inequivalent spaces are inequivalent. One way to distinguish between inequivalent topological spaces is to find some topological property that one has and the other hasn’t. For example, any closed curve on a sphere divides it into two pieces, but on the torus there exists closed curves which do not cut it into separate pieces. As another example, the real line is not topologically equivalent to any closed bounded interval, as the real line is non-compact.. the properties : closed curve, connectedness, disconnectedness, compactness are some familiar topological invariants.