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

In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding ιn:K[[T1,...,Tn]]K[[X,Y]]. Now let us define the infinitely many variables formal power series as follows:

K[[T1,...,T]]:=limn1K[[T1,...,Tn]].

For example, i=1Ti=T1+T2+T3+...K[[T1,...,T]].

Then I would like to ask

Q. Can K[[T1,...,T]] be embedded into K[[X,Y]]?

That is, does the embedding ι:K[[T1,...,T]]K[[X,Y]] exist?

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

Answer

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 (ti)i1 of elements of K[[X]], which is algebraically independent over K. The continuous morphism of topological K-algebra from K[[(Ti)i1]] to K[[X,Y]] given by Titi(X)Yi is injective : if we give weight i to the variable Ti, then any element F of K[[(Ti)i1]] can be expanded as a convergent series
F=i0Fi(T1,,Ti),
where Fi is a (weighted) homogeneous polynomial of degree i. The image of F in K[[X,Y]] is given by the convergent series
i0Fi(t1(X),,ti(X))Yi,
which vanishes iff Fi(t1(X),,ti(X))=0 for each i, iff Fi=0 for each i.

Attribution
Source : Link , Question Author : Pierre MATSUMI , Answer Author : js21

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