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∞]]:=lim←n≥1K[[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)i≥1 of elements of K[[X]], which is algebraically independent over K. The continuous morphism of topological K-algebra from K[[(Ti)i≥1]] to K[[X,Y]] given by Ti↦ti(X)Yi is injective : if we give weight i to the variable Ti, then any element F of K[[(Ti)i≥1]] can be expanded as a convergent series

F=∑i≥0Fi(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

∑i≥0Fi(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*