Topology defined by a fundamental system of neighbourhoods of zero in a topological group What are the open sets in a topology specified by a fundamental system of neighbourhoods of $0$ of a topolgical group? Also, how is this topology unique. I searched this online, but the books I found only mention these statements, without specifying what the open sets are and why the topology is unique. Any help will be appreciated.
 A: Basically, because to define a topology we need to define a local base at every point in a consistent way. In a topological group $G$, for every $x$ and $y$ we have a homeomorphism of $G$ that maps $x$ to $y$, just use $h(z) = y*x^{-1}*z$. So we can transport a neighbourhood base of $e$ (the identity of $G$) to any other point of $G$, using such $h$. One then checks this is a consistent assignment and so determines the topology on $G$.
A: I am not very familiar with topological groups, but this is what I know from general topology.
You have neighborhood base at every point $x \in G$ (obtained as Henno Brandsma described). Now open sets are obtained uniquely in this manner:
set $U$ is open if and only if for every point $x \in U$ there exists an element $V$ of neighborhood base at $x$ such that $V \subset U$.
Verification of this fact is left as an exercise to the reader :) (and I don't know of a good citation at the moment).
PS: See also the discussion in here: How can you construct a topology from a fundamental system of neighborhoods?
