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It is not assumed that if two Ui's intersect then their intersection is connected, as is the case for disks in the plane. Note, however, that if the Ui were chosen to be small geodesic disks for a conformal Riemannian metric, locally of the form ds2 = f(z) |dz|2, then any non-empty intersection of finitely many Ui would be geodesically convex and hence connected; see do Carmo 1976, pp. 303–305. do Carmo, Manfredo Perdigão (1976). Differential geometry of curves and surfaces. Englewood Cliffs, N.J.: Prentice-Hall. ISBN9780132125895. 2016 reprint