The definition of a topological group says that it is a group where the inversion and multiplication are continous, but what exactly are the open sets in this topological space? and how we can show that once the inversion and multiplication are continuous, it satisfies the axioms of a topological space?
Could anyone please explain the idea and maybe the intuition of such an object? Thanks!
On
This might be a late answer, but I think there is a necessity to explain things in a more intuitive way.
A topological group is a group (as a set) endowed with some topological structure (think of this structure as the collection of those open sets). The first chapter of the any topology book will tell you that a set could be endowed with any topology (e.g., trivial topology, discrete topology) and it is also the case for a group.
However, when dealing with a group, the choice of topology cannot be that arbitrary, as the topology must satisfy two properties: multiplication continuous and inverse continuous. In other words, any topology that satisfy these two properties could serve as a group topology.
So, when you ask what are the open sets of a group topology, it could be anything; and vice versa, an open set in one group topology could be open/closed in another group topology.
The key point that: The topology (open sets) are not uniquely determined by the group itself.
Hope this clarifies a bit.
A topological group, $G$, is a topological space which is also a group such that the group operations of product:
$G\times G\to G:(x,y)\mapsto xy $
and taking inverses:
$ G\to G:x\mapsto x^{-1}$
are continuous. Here $G × G $ is viewed as a topological space with the product topology.