Let $M$ be a [topological manifold](/page/Topological%20Manifold), and let $(U,\varphi)$ and $(V,\psi)$ be charts on $M$, so that $U,V\subset M$ are [open sets](/page/Open%20Set), $\varphi:U\to \varphi(U)$ and $\psi:V\to \psi(V)$ are [homeomorphisms](/page/Homeomorphism) onto their chart images, and the chart images carry the [subspace topologies](/page/Subspace%20Topology) inherited from Euclidean space. Then the transition map $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$ is a homeomorphism, where $\varphi^{-1}$ denotes the inverse map $\varphi(U)\to U$ restricted to $\varphi(U\cap V)$.