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My textbook says that "a chart or coordinate system consists of a subset [itex]U[/itex] of a set [itex]M[/itex], along with a one-to-one map [itex]\phi :U\rightarrow\mathbf{R}^n[/itex], such that the image [itex]\phi(U)[/itex] is open in [itex]\mathbf{R}^n[/itex]."
What's the motivation for demanding that the image of [itex]U[/itex] under [itex]\phi[/itex] be open?
What's the motivation for demanding that the image of [itex]U[/itex] under [itex]\phi[/itex] be open?