I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of infinite extensions and here I get into some trouble. I am not familiar with topology at all. I have tried reading a bit myself but I am only aware of the very basic definitions. To be more specific, I don't have any "working experience" with topology - I know many of you will say this is unacceptable, but my University does not offer any courses on topology at all.

In particular, I can not understand what the phrase "endowed with the [some topology]". What does that really mean? Or phrases of the form "[some topology A] agrees with [some topology B]". This sort of situation appears in the definition of the topological Galois group of an infinite extension of fields. If I understand correctly, there is something called "the Krull topology" associated to the infinite Galois group, which is the inverse limit of Galois groups of finite extensions. I don't understand how this works.

Is there somewhere I can learn how to deal/understand the idea of topology in the context needed for Infinite Galois theory, or do I have to learn topology first from scratch. Thanks in advance for any help.

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    $\begingroup$ You can use the internet. For example, look up the definition of a topology on Wikipedia. Skim Topology Without Tears (available freely online) and pay attention to the definitions. $\endgroup$ – blue May 1 '14 at 0:00
  1. The same set can have different topologies on it. For a set $X$, a topology on $X$ is a set $\tau$ of subsets of $X$ such that $\cup_{i \in I} U_i \in \tau$ when $U_i \in \tau$, $\cap_{ i = 1}^n U_i \in \tau$ when $U_i \in \tau$, and $\emptyset, X \in \tau$. "$X$ is endowed with the topology $\tau$" is just another way of saying "$(X, \tau)$ is a topological space, and we are going to write $X$ to refer to that topological space". This is a common abuse of notation, where the symbol $X$ refers both to the set $X$ and to the topological space $(X, \tau)$. There is no confusion in context.

  2. If people say two topologies agree and they are topologies on the same set then they just mean that they are the same topology. Otherwise they may mean that there is a homeomorphism between topological spaces.

Understanding the topology one puts on an inverse limit may be hard if you have not seen topology before. I don't mind Munkres's book. I don't have a favorite though.

Anyways, to put a topology on the inverse limit, we view the finite galois groups $\{ G_i \}$ as discrete topological spaces (all subsets open and closed) and take the inverse limit in the category of topological spaces. The inverse limit $\text{lim } G_i$ is a subspace of the product $\prod G_i$. To get a topology on the inverse limit $\text{lim} G_i$ we first put the product topology on $\prod G_i$, and then the subspace topology on $\text{lim } G_i \subset \prod G_i$.

So prerequisites for understanding this are:

  1. You might want to understand inverse limits.

  2. There is a construction on topological spaces called the product topology. It satisfies the universal property of product in the category of topological spaces.

  3. You have to know the subspace topology.

It may also be good to understand profinite sets, and profinite spaces. This gets into stone duality, but that isn't really necessary to know here.


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