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In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the infinitely many variables formal power series as follows$\colon$

$K[[T_1,...,T_\infty]] \colon= \,\underset{n \geq 1}{\varprojlim} K[[T_1,...,T_n]]$.

For example, $\sum^{\infty}_{i=1} T_i = T_1 + T_2 + T_3\, + \,... \in K[[T_1,...,T_\infty]]$.

Then I would like to ask

Q. Can $K[[T_1,...,T_∞]]$ be embedded into $K[[X,Y]] \,?$

That is, does the embedding $\iota_{\infty} \colon K[[T_1,...,T_\infty]] \hookrightarrow K[[X,Y]]$ exist?

  Embeddings are meant to be continuous injective $K$-algebra homomorphisms.

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  • $\begingroup$ Could you link to the "question 95"? I found mathoverflow.net/questions/95 but it's not clear to me how it's related. Second, you could say what you mean by "embedding": as rings? as $K$-algebras? as topological $K$-algebras? $\endgroup$
    – YCor
    Jun 20, 2016 at 12:15
  • $\begingroup$ @YCor This answer of this question currently has 95 upvotes. It looks like the morphism needs to be continuous and injective. $\endgroup$
    – Yuzhou Gu
    Jun 20, 2016 at 12:58
  • $\begingroup$ I mean that the embedding is just injective homomorphism $\iota_{\infty}$. $\endgroup$ Jun 20, 2016 at 13:53
  • $\begingroup$ Sorry. The number is Question 98. One can find it in the following:mathoverflow.net/questions/23478/… $\endgroup$ Jun 20, 2016 at 14:00
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    $\begingroup$ It has no sense to call it question 98, unless you want to confuse everyone. This is Question 23478, and you refer to one of the answers, which at this very time has 98 upvotes... $\endgroup$
    – YCor
    Jun 20, 2016 at 14:37

1 Answer 1

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The field $K((X))$ has infinite transcendence degree over $K$ (if $K$ is at most countable, this just follows from a cardinality argument). Thus we can find a countable family $(t_i)_{i \geq 1}$ of elements of $K[[X]]$, which is algebraically independent over $K$. The continuous morphism of topological $K$-algebra from $K[[(T_i)_{i \geq 1}]]$ to $K[[X,Y]]$ given by $T_i \mapsto t_i(X)Y^i$ is injective : if we give weight $i$ to the variable $T_i$, then any element $F$ of $K[[(T_i)_{i \geq 1}]]$ can be expanded as a convergent series $$ F = \sum_{i \geq 0} F_i(T_1,\dots,T_i), $$ where $F_i$ is a (weighted) homogeneous polynomial of degree $i$. The image of $F$ in $K[[X,Y]]$ is given by the convergent series $$ \sum_{i \geq 0} F_i(t_1(X),\dots,t_i(X)) Y^i, $$ which vanishes iff $F_i(t_1(X),\dots,t_i(X)) = 0$ for each $i$, iff $F_i =0$ for each $i$.

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