Manifold

Charts and transition maps

A K-chart at p is a homeomorphism from an open neighbourhood of p to K. Usually, K is taken to be an open subset of Rn. If at p there are two charts, a K1-chart and a K2-chart, then by restricting them to the intersection of their domains we can compose the inverse of one with the other to form a transition map from an open subset of K1 to an open subset of K2 -- in other words, from an open subset of Rn to another open subset. All transition maps are continuous (as compositions and restrictions of continuous maps), and since the inverse of a transition map is also a transition map (by inverting the roles of K1 and K2), all transition maps are homeomorphisms. The definition of a manifold implies that for every p in the manifold, there exists a chart.

Related Topics:
Homeomorphism - Neighbourhood - Subset - Transition map - Continuous

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