- #1

yifli

- 70

- 0

Let M be a manifold with atlas A. A sequence of points [itex]\{x_i \in M\}[/itex] converges to [itex]x\in M[/itex] if

- there exists a chart [itex](U_i,\phi_i)[/itex] with an integer [itex]N[/itex] such that [itex]x\in U_i[/itex] and for all [itex]k>N,x_i\in U_i[/itex]
- [itex]\phi_i(x_k)_{k>N} \rightarrow \phi_i(x)[/itex]

Given a sphere (2-manifold) centered at origin and a sequence of points converging to the north pole. The atlas of this sphere contains two charts, which projects all the points on the lower semi-sphere [itex]U_1[/itex] (resp., upper semi-sphere [itex]U_2[/itex]) from the south pole (resp., north pole) to the x-y plane, i.e.,

[tex]\phi_1(x_1,x_2,x_3)=\langle \frac{x_1}{1+x_3}, \frac{x_2}{1+x_3} \rangle[/tex]

[tex]\phi_2(x_1,x_2,x_3)=\langle \frac{x_1}{1-x_3}, \frac{x_2}{1-x_3} \rangle[/tex]

Since the sequence of points converge to the north pole, we can find an N such that [itex]x_k \in U_2, k > N[/itex]; however, [itex]\phi_2(x_k) \rightarrow \infty[/itex], which means this sequence is not convergent. How come?