I'm starting to study Infinite Galois Theory and its relation with Profinite Groups, but I'm having troubles with basic definitions.
Definition 1. Let $K/F$ a Galois extension. Write $$\mathcal{F} = \{L \mid L \text{ is a subfield of }K\text{ s.t. } L/F\text{ is a finite Galois extension}\}.$$ We define a topology in $\mathrm{Gak}(K/F)$ by taking as a base of open neighborhoods of $1$ the family of subgroups $$\mathcal{N} = \{\mathrm{Gal}(K/L) \mid L \in \mathcal{F}\}.$$
Definition 2. The Krull Topology on $\mathrm{Gal}(K/F)$ is defined as follows: A subset $X$ of $\mathrm{Gal}(K/F)$ is open if is empty or $X = \bigcup_{i}g_{i}N_{i}$ for some $g_{i} \in G$ and $N_{i} \in \mathcal{N}$.
We can show that, according to definition 2, the basis of Krull Topology is $$\{gN \mid g \in G, N \in \mathcal{N}\},$$ but I just can see that taking $g = 1$, the definition 2 becomes the definition 1.
Questions:
- How can I see that 1 and 2 defines the same topology?
Maybe it's a stupid question, but
- Why defining the open neighborhoods of $1$ it's enough to generate a topology?
I can see why the definition 2 works. Unfortunately, my book uses the definition 1. Thus, I would like to understand why that definition works and is equivalent to the second.
Thank you for any help!
The thing is that we don't just define any topology, we define a group topology. This means that we want a topology that makes $G=Gal(K/F)$ a topological group. A topological group is homogeneous as a topological space, so as far as the topology goes, the points are all indistinguishable. This is because if $g\in G$ is any point, then the map $\lambda_g: G\to G$ given by $x\mapsto gx$ is a homeomorphism that sends $1$ to $g$.
So if you want to describe a basis of open neighborhoods of any point, you only need to describe it for $1$. It is then implied that if $\{U_i\}$ is a basis of neighborhoods for $1$, then $\{gU_i\}$ is taken as a basis of neighborhoods for $g$.
With this in mind, you can check that the two descriptions are the same.