I was wondering about if topological quotients and group quotients always agree for a topological group. By this, I mean if G is a topological group, G/~ is a topological quotient space, G/N is a group quotient where N is a normal subgroup, and the equivalence classes generated by ~ are the same as the cosets generated by N, then will G/~ and G/N be isomorphic? One example is where ~ is defined as a~b if a and b differ by an integer. Then R/~ sometimes shorthand written as R/Z as a quotient topology is the circle S. Also, Z is a normal subgroup of R so one can take R/Z as a group quotient which is the circle group S. In both cases we can write R/Z=S, but the meanings are quite different depending on if this is a topological quotient or group quotient. I was wondering if this is a property in general or possibly a special case of using “nice” sets like R and Z.
I was also wondering if this might be a case of a more general property. The topological quotient can be informally thought of as gluing 0 and 1 and making sure an open set can go into 0 and come out at 1 while the group quotient can be informally thought of as making sure an addition of an element can go into 0 and come out at 1. In both cases the quotient seems to glue the points 0 and 1 and also glue the structure there whether a topological structure or group structure. This seems to be formalized in category theory by the idea of a quotient category, but I was wondering if in general quotients for different types of categories will agree like the do for R/Z=S.
Let $G$ be a topological group, $N\subset G$ a normal subgroup. One has an abstract group $G/N$ of course, and one can apply the quotient topology to it. The thing to check is that $G/N$ endowed with this topology becomes a topological group. It is it turns out!
See theorem 4.15 in
https://www.lakeheadu.ca/sites/default/files/uploads/77/images/Spivak%20Dylan.pdf