I'm stuck on this example in Boto v. Querenburgs "Mengentheoretische Topologie" and I would really appreciate some insight from our more topologically savvy friends on here. =)
Let $I$ be an ordered set, and let $(X_j, \mathcal T_j)_{j \in I}$ be a family of topological spaces, such that for $j
$X_j \subset X_k, \quad \mathcal T_j = \mathcal T_k|X_j$
i.e. the topology on $X_j$ is induced by the injection $i_{jk}: X_j \hookrightarrow X_k$ from the topology on $X_k$.
On $X = \bigcup_{j\in I} X_j$ let $\mathcal T \; $ be the final topology with respect to $(i_j: X_j \hookrightarrow X)_{j\in I}$; This is called the weak topology on $X$.
Example: Let $X_n = \mathbb R^n$, and let $X = \mathbb R^\infty$. Then a sequence $(x_k = (x_{k1}, x_{k2}, \dots ))_{k\in \mathbb N}$ converges to $x = (x_1, x_2, \dots )$ iff for any fixed $n$ the sequence $(x_{kn})_{k \in \mathbb N}$ converges to $x_n$.
Now I don't see why the last statement should be true.
First off: To get a feel for this new kind of topology, I tried comparing it to other topologies on $\mathbb R^\infty$ known to me (and I think in the following already I must be making a mistake...)
Suppose $U = \prod_{n \in \mathbb N} U_n$ is open in the box topology on $\mathbb R^\infty$, i.e. $U_n \subset \mathbb R$ is open for all $n$. Then I think $U$ is also open in the weak topology: (I suppose $\mathbb R^j$ should be identified with $\mathbb R^j \times 0 \times 0 \times \dots \subset \mathbb R^\infty$, right?)
But then $i_j^{-1}(U) = \emptyset$ if $0 \notin U_n$ for some $n>j$ and $i_j^{-1}(U) = U_1 \times \dots \times U_j$ otherwise. Both of which are open in $\mathbb R^j$, thus $U$ should be open in the final topology w.r.t. the inclusions $i_j$.
Now consider the sequence
$x_j = \underset{\text{j-th component}}{(0, \dots, 0,\underbrace{1}, 0, 0, \dots)}$
Clearly $x_j$ converges to $(0, 0,\dots)$ componentwise, but $x_j$ does not converge in the box topology, hence neither in the finer topology introduced in the example (the weak topology).
So where is the above argument wrong? What am I not understanding correctly about this topology?
Many, many thanks in advance for any useful comments and answers.
Regards,
S.L.