In this paper, they defined the projective Lie-group and pro-lie group to be:

Projective Lie Group: A topological group is called projective limit of Lie groups, if there is a projective system $$\{f_{jk}:G_k \rightarrow G_j\mid j\leq k,(j,k)\in J \times J\}$$ for a directed index set J and for finite-dimensional Lie groups $G_j$ and if$$G=\lim_{j\in J}G_j=\{(g_j)_{j\in J}\in\prod_{j\in J}G_j:(\forall j\leq k)g_j=f_{jk}(g_k)\}$$

Pro-Lie Group: A pro-lie group is a complete topological group and every identity neighbor hood contains a normal subgroup N s.t $G/N$ is a finite dimensional Lie group and every intersection of every two such normal sub groups contains a third of the same type. < They showed that Pro-Lie Group is a projective Lie group in the following way:

let $\mathcal{N}(G)$ denote the filter basis of all $N$ such that $G/N$ is a finite dimensional Lie group. Then the natural quotient maps $G/N → G/M$ for $M ⊇ N$ in $\mathcal{N}(G)$ form a projective system such that $G\cong \lim_{N∈\mathcal{N}(G)} G/N$.

Question: Why is $G$ isomorphic to the projective limit? i.e. why $G\cong \lim_{N∈\mathcal{N}(G)} G/N$. The left-hand side is a infinite direct product while the former is just one element from $G$.


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