In nlab:
https://ncatlab.org/nlab/show/category+of+G-sets
It is said that for every topological group $G$ there is a dicrete version of it $G^\delta$ in an obvious way.
Somebody can help me with a reference. I expect that this group will be discrete. What is the construction of it?
Take, as a topology on $G$, the topology in which every set is open. That's it!
One consequence: the connected component of the idenitty is just the identity.
Also, any map from $G$ to anything is continuous, so to check that something's a topological group homomorphism, you just have to check "homomorphism" -- you get "continuous" for free.