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Let $M$ be a manifold with an atlas $\mathbb{a}$. A sequence of points $\{x_i \in M\}$ converges to $x\in M$ if

  • there exists a chart $(U_i,\phi_i) \in \mathbb{a}$ with an integer $N_i$ such that $x\in U_i$ and for all $k>N_i,x_k \in U_i$
  • $\phi_i(x_k)_{k>N_i}$ converges to $\phi_i(x)$

I don't think the above definition is consistent. For example, given a 2-manifold sphere and a sequence of points converging to the north pole. The atlas of the manifold consists of two charts, which projects the points on the lower semi-sphere (resp., upper semi-sphere) from the south pole (resp., north pole) , i.e.,

  • $\displaystyle \phi_1(x_1,x_2,x_3) = \left\langle \frac{x_1}{1-x_3},\frac{x_2}{1-x_3}\right\rangle$
  • $\displaystyle \phi_2(x_1,x_2,x_3) = \left\langle \frac{x_1}{1+x_3},\frac{x_2}{1+x_3}\right\rangle$
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Are you asking whether the definition is consistent? For confirmation/refutation of your thoughts? It helps to ask explicit questions...also: it seems you haven't accepted any answers to your previous questions. If answers aren't clear to you, ask for clarification. Otherwise, it is considered "good manners" to accept (one of/the) answer which you found helpful/most helpful for each question: you can do that by clicking on the grey "check mark" next to a given answer. You can also vote up any and all answers you found helpful (click on upward arrow). –  amWhy Jun 25 '11 at 0:04
    
@amWhy: the $\lt$ was part of a scalar product. Please use \langle a, b \rangle for such cases. Compare: $\lt a,b \gt = \langle a, b \rangle$, obtained by \lt a,b \gt = \langle a, b \rangle –  t.b. Jun 25 '11 at 0:12
    
@Theo I assumed the user was trying to express less than or equal too...clearly didn't read the problem through...Thanks for undoing my correction and editing OP's incorrect use of < > instead of the angle bracket notation (knowing what was meant... –  amWhy Jun 25 '11 at 0:19

1 Answer 1

up vote 1 down vote accepted

I am assuming that you are asking regarding the equivalence between "convergence in the topological sense" and "convergence in the manifold sense". More precisely:

(1) [Convergence in the "Topological Sense"] A sequence $\{x_n\}$ of points in a topological space $X$ is said to converge to $x\in X$ if to each neighborhood $U$ of $x$, there exists a natural number $N$ such that $n\geq N$ implies $x_n\in U$.

(2) [Convergence in the "Manifold Sense"] The definition you have given at the beginning of your answer.

Proof. (1)=>(2): If $x\in M$ and if $\{x_n\}$ is a sequence converging to $x$ in the sense of (1), then we wish to prove that $\{x_n\}$ converges to $x$ in the sense of (2). We certainly know that there is a chart $(U,\phi)$ such that $x\in U$. Choose a natural number $N$ such that $n\geq N$ implies $x_n\in U$. We wish to show that the sequence $\{\phi(x_n)\}_{n\geq N}$ converges to $\phi(x)$. This is a consequence of the following exercise:

Exercise 1: Prove that if $\{x_n\}$ is a sequence of points in a topological space $X$ such that $x_n\to x$ and if $f:X\to Y$ is a continuous map of topological spaces, then $\phi(x_n)\to \phi(x)$.

Therefore $x_n\to x$ in the sense of (2).

(2)=>(1) Conversely, suppose $x_n\to x$ in the sense of (2); we wish to prove that $x_n\to x$ in the sense of (1). Choose a neighborhood $V$ of $x$. We know that there exists a chart $(U,\phi)$ such that $x\in U$. Therefore, there exists (by (2)) a natural number $N$ such that $x_n\in U$ for all $n\geq N$. We also know that the sequence $\{\phi(x_n)\}_{n\geq N}$ converges to $\phi(x)$. However, $\phi$ is a homeomorphism and therefore its inverse is continuous. In particular, we can apply Exercise 1 to $\phi^{-1}$ to conclude that $\{\phi^{-1}(\phi(x_n))\}_{n\geq N}\to \phi^{-1}(\phi(x))$ in $U$, i.e., $\{x_n\}_{n\geq N}\to x$ in $U$. Therefore, there exists $M\geq N$ such that $n\geq M$ implies $x_n\in U\cap V\subseteq V$ and $x_n\to x$ in the sense of (1).

Exercise 2: Prove the following generalization of a result noted in the implication (2)=>(1) in the above proof: if $f:X\to Y$ is a homeomorphism of topological spaces and if $\{x_n\}$ is a sequence in $X$ such that $f(x_n)\to f(x)$ for some $x\in X$, then $x_n\to x$.

Exercise 3: Does there exist an example of a continuous map $f:X\to Y$ between topological spaces and a sequence $\{x_n\}$ of points of $X$ such that $f(x_n)\to f(x)$ but that $x_n\not\to x$? (Prove or give a counterexample.)

Exercise 4: Let $f:X\to Y$ be a map such that whenever $\{x_n\}$ is a sequence of points with $x_n\to x$ for some $x\in X$, we also have $f(x_n)\to f(x)$. If $X$ and $Y$ are first countable topological spaces, then prove that $f$ is continuous.

Exercise 5: Is the result of Exercise 4 true if the word "first countable" is deleted every time it appears?

Exercise 6: You might wish to study the concept of a net in point-set topology. This concept allows us to define a type of "generalized sequence" that is very useful in the context of topological spaces that are not first countable. I leave it as an exercise to search up the concept of a net and learn more about it (if you have not already).

I hope this helps!

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