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

Question

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.

Answer 1

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 $i$.