# Definition A topological invariant is a property of a [[Topological Space]] that is invariant under [[Homeomorphism|homeomorphisms]]. # Examples Examples include the number of [[Connectedness|connected]] components of the space, an algebraic structure such as a [[group]] or a [[ring]] which is constructed out of the space, [[Connectedness]], [[Compactness]], [[Hausdorff Space|Hausdorff property]]. Arguably the most famous topological invariant is the [[genus]] of a closed surface, given by the [[Gaus-Bonnet theorem]]. # Physics In physics, we simply say that topological invariant is a quantity that does not change under a continuous or [[Adiabatic]] transformation. Most topological invariants in physics are integrals of *geometric* quantities. ## Examples In physics, we have e.g. the [[Chern number]].