 Topological transition map

For two coordinate charts in a topological atlas, the topological transition map is the homeomorphismBoth domains are open sets in Euclidean space. Function composition gives $g_{ij}=g_{ji}^{-1}$ and $g_{ki}=g_{kj}\circ g_{ji}$ wherever all three coordinate charts overlap. These identities express consistency of the coordinate descriptions; imposing smooth functions as the coordinate changes gives a smooth atlas.