Krull Topology

**ThePerfectHacker** · August 3rd 2008, 03:58 PM

Let $K/F$ be an algebraic (infinite) Galois extension with group $G$ .
We can define the Krull topology on the group $G$ .
We wish to show this topology is totally disconnected.
Let $\phi_1,\phi_2 \in X$ some subset of $G$ .
Then there exists a subgroup $N\subseteq G$ of the form $\text{Gal}(K/E)$ where $E/F$ is a finite Galois extension with $\phi_2\not \in \phi_1 N$ (this is just a property on how this topology is defined).
Then, $X = (\phi_1 N \cap X) \cup ((G-\phi_1 N) \cap X)$ .
Thus, $X$ is a union of two non-empty open sets.
My question is why are those sets open?

**NonCommAlg** · August 4th 2008, 08:24 AM

Originally Posted by ThePerfectHacker

Let $K/F$ be an algebraic (infinite) Galois extension with group $G$ .
We can define the Krull topology on the group $G$ .
We wish to show this topology is totally disconnected.
Let $\phi_1,\phi_2 \in X$ some subset of $G$ .
Then there exists a subgroup $N\subseteq G$ of the form $\text{Gal}(K/E)$ where $E/F$ is a finite Galois extension with $\phi_2\not \in \phi_1 N$ (this is just a property on how this topology is defined).
Then, $X = (\phi_1 N \cap X) \cup ((G-\phi_1 N) \cap X)$ .
Thus, $X$ is a union of two non-empty open sets.
My question is why are those sets open?

the set $\{g\text{Gal}(K/E): \ g \in G, \ F \subseteq E \subseteq K \ \text{such \ that} \ [E:F] < \infty \}$ is a basis for the Krull topology. so $\phi_1N$ is open in G.

$G-\phi_1N$ is a union of (finitely many) cosets of N and therefore it's also open in G. thus $\phi_1N \cap X$ and $(G-\phi_1N) \cap X$ are

open in $X.$ (here $X$ is considered with the subspace topology inherited from G.)

**ThePerfectHacker** · August 4th 2008, 07:20 PM

Thank you. I did not notice that we were working in the topology of X rather than G.
(Because it does not need to be open in G).

This is Mine 1

3

th Post!!!

**ThePerfectHacker** · August 9th 2008, 06:21 PM

Here is a related question I was thinking about.
Let $\sigma$ be any automorphism.
Let $H$ be a subgroup of $\text{Gal}(K/F)$ .
Let $C$ be a closed set such that $H\subseteq C$ .
If for any open set $\sigma N$ intersects $H$ prove $\sigma \in C$ .

**NonCommAlg** · August 9th 2008, 07:24 PM

Originally Posted by ThePerfectHacker

Here is a related question I was thinking about.
Let $\sigma$ be any automorphism.
Let $H$ be a subgroup of $\text{Gal}(K/F)$ .
Let $C$ be a closed set such that $H\subseteq C$ .
If for any open set $\sigma N$ intersects $H$ prove $\sigma \in C$ .

so for some $\sigma \in \text{Gal}(K/F)$ we know that $\sigma N \cap H \neq \emptyset$ for all N which are of the form Gal(K/E), for some finite extension E/F. is this what you're assuming? if so, then i don't think we'll need

H to be a subgroup. that's my first thought, but i guess i need a few minutes to think about it!

**ThePerfectHacker** · August 9th 2008, 07:42 PM

Originally Posted by NonCommAlg

so for some $\sigma \in \text{Gal}(K/F)$ we know that $\sigma N \cap H \neq \emptyset$ for all N which are of the form Gal(K/E), for some finite extension E/F. is this what you're assuming? if so, then i don't think we'll need

H to be a subgroup. that's my first thought, but i guess i need a few minutes to think about it!

Let $H$ be a subgroup of the Galois group. Let $H_0 = \text{Gal}(K/K^H)$ . I wish to prove that $\bar H = H_0$ (the topological closure).
It is sufficient to prove that $H_0$ is closed and $H_0 \subseteq \bar H$ .
The closed part is not the problem. It is the inclusion which is giving me a problem.

Thus, far I know that if $N$ is of form $\text{Gal}(K/E)$ where $E/F$ is finite Galois and $\sigma \in H$ then $\sigma N \cap H \not = \emptyset$ .
And this is somehow supposed to prove that therefore $\sigma \in \bar H$ .
That is what I do not see.

Since $\bar H$ is the intersection of all closed sets which contain $H$ it is sufficient to prove that if $C$ is a closed set which contains $H$ and $\sigma N$ intersects $H$ for all $N$ (those "special" groups) then $\sigma \in C$ and therefore $\sigma$ is in the intersection. This was the theorem I assumed did the trick - and which is why I asked my question above.

**NonCommAlg** · August 9th 2008, 08:00 PM

Originally Posted by ThePerfectHacker

Here is a related question I was thinking about.
Let $\sigma$ be any automorphism.
Let $H$ be a subgroup of $\text{Gal}(K/F)$ .
Let $C$ be a closed set such that $H\subseteq C$ .
If for any open set $\sigma N$ intersects $H$ prove $\sigma \in C$ .

suppose $\sigma \notin C.$ then since C is closed, there exists an open set $\alpha N$ which contains $\sigma$ and $\alpha N \cap C = \emptyset.$ but since $\sigma \in \alpha N,$ we must have $\alpha N = \sigma N.$

so $\sigma N \cap C = \emptyset,$ and hence $\sigma N \cap H = \emptyset,$ because $H \subseteq C.$ contradiction! as you see H doesn't need to be a subgroup!

**ThePerfectHacker** · August 12th 2008, 02:17 PM

I understand what you did but I was thinking.
Is the theorem "a set is open if and only if it contains all its interior points" still true in general topology?
One direction is obvious but I do not think the other direction needs to holds.

The above theorem is of course true for the Krull Topology basically by definition.
And I know it is true for metric spaces - in fact it can be regarded as a definition of 'open'.
But I cannot imagine it working in general topology.

(I never studied topology before this is why I keep on getting these basic questions).

**NonCommAlg** · August 12th 2008, 02:33 PM

Originally Posted by ThePerfectHacker

Is the theorem "a set is open if and only if it contains all its interior points" still true in general topology?

it's very easy to see that the interior of a set in a topological space is the largest open subset of that set. so a set is open if and only if it's equal to the set of its interior points.

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