Does product of Galois groups equal to the Galois group of corresponding fields intersection?

Question

Let $k$ be a field, $\bar{k}/k$ be a Galois extension with $G=Gal(\bar{k}/k)$ be an Abelian group(may be infinite). If $K,L$ are intermediate fields, denote $G_K=Gal(\bar{k}/K), G_L=Gal(\bar{k}/L)$. Is it true $G_K G_L=G_{L\cap K}$? Generally, is it true $\langle G_{K_i}\mid i\in I\rangle=G_{\cap K_i}$?(I guess it is true when thinking of $G_{\cap K_i}$ being topologically generated by$G_{K_i}$, which I don't know what does that exactly mean.)

Answer 1

$G_K G_L=G_{L\cap K}$ is a consequence of Galois correspondence: $G_K G_L$ is the smallest group containing both $G_K$ and $G_L$, so it fixes the largest field contained in both $K$ and $L$, i.e. $K\cap L$. In the infinite degree case, Galois correspondence is between fields and closed subgroups; you are thus right, you need to take the closure of $\prod G_{K_i}$.