More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?


Question: Examples of RESTful Web Services for Browser Games
Answer: I think the network protocol is really irrelevant, each game uses its own protocol, REST or any other standard is only important if you need to interoperate with third party software - which game servers mostly do not.

For example, if you published the API, using REST might be a good idea. But otherwise, I don't see any reason to do so.