Difference between $ \mathbf R^{\infty}$ and $ \mathbf R^w$ In Munkres's topolgy, they define $ \mathbf R^w$ as infinite cartesian product of $R$ with itself.
But I cant tell the difference between $  \mathbf R^{\infty}$ and $  \mathbf R^w$.
Does it matter what 'infinity' means between countably infinite and uncountable?
 And they say  $  \mathbf R^{\infty}$ is the union of the spaces $\mathbf R^{~n}$ whose elements are $(x_1,x_2,...)$ with $x_i=0$ if $i>n$.
But I cant understand this. For example, $(0,1,0,1,0,..)$ does not belong to any $\mathbf R^{~n}$. 
 A: The infinity here is always countable. What you've noticed is the difference bettween an infinite direct sum and an infinite direct (cartesian) product of vector spaces. While direct sum and direct product of vector spaces are isomorphic for finite number of summands/factors, it's no longer true if their number is infinite.
You've correctly identified the difference: in the direct product we can have arbitrary sequences, while in the direct sum we require that from some point all the elements of the sequence are equal to $0$.
The reason why such difference exists comes from how direct product and direct sum are defined. 


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*The direct product of vector spaces $(A_i)_{i\in I}$, denoted as $\prod_{i\in I} A_i$, is defined as a vector space equipped with projections $\pi_k : \prod_{i\in I} A_i \rightarrow A_k$ such that for any vector space $B$ and any linear maps $f_k : B \rightarrow A_k$ there exist a linear map $g: B \rightarrow \prod_{i\in I} A_i$ such that $\forall k\in I: \pi_k \circ g = f_k$.


If we have a countable case we can represent direct product with infinite sequences, we have $\pi_k(a_1,a_2,a_3,\dots) = a_k$ and for given $f_k: B\rightarrow A_k$ we can find that $g(b) = \big(f_1(b),f_2(b),f_3(b),\dots\big)$. Since $f_k$ are arbitrary, we need to allow all sequences.


*The direct sum of vector spaces $(A_i)_{i\in I}$, denoted as $\bigoplus_{i\in I} A_i$, is defined as a vector space equipped with coprojections $\alpha_k : A_k \rightarrow \bigoplus_{i\in I} A_i $ such that for any vector space $B$ and any linear maps $f_k : A_k \rightarrow B$ there exist a linear map $g: \bigoplus_{i\in I} A_i \rightarrow B$ such that $\forall k\in I: g\circ \alpha_k = f_k$.


If we have a countable case we can represent direct product with infinite sequences, and we have $\alpha_k(a) = \big(0,0,\dots,0,a,0,\dots\big)$, where $a$ has been inserted at k-th place. For given $f_k: A_k\rightarrow B$, we can then find $g(a_1,a_2,\dots) = \sum_{k}f_k(a_k)$. To guarantee that this sum exist  for arbitrary $f_k$, we need to assume that only finite number of $a_k$ is non-zero, which means that from some point they all need to be $0$.
