I am currently working with the book "Fuchsian groups", by Svetlana Katok and am trying to solve a few of the provided exercises.
I understand $PSL(2,\mathbb{R})$ can be represented as the quotient group $SL(2,\mathbb{R})/(±Id)$ and $PSL(2,\mathbb{R})$ is a topological group endowed with a quotient topology. Also it is clear to me how to show that the multiplication and inverses are continuous for $SL(2,\mathbb{R})$ since it is a subset of $\mathbb{R}^4$.
However, how would one now show that the group multiplication and inverse are continuous with respect to the topology on $PSL(2,\mathbb{R})$?
Let $G$ be a topological group, $N$ a normal (topological) subgroup. We want to show that the quotient group operation $m: G/N \times G/N \to G/N$ is continuous.
We can precompose this map with the product of the canonical projection maps $p: G \times G \to G/N \times G/N$.We can show that this is open and continuous, i.e. a quotient map. We get $mp: G \times G \to G/N$ and now we only need to prove that this is continuous:
But this map is equal to the one obtained by composing the group operation map $G\ \times G \to G$ with the projection $G \to G/N$. This map is evidently continuous, so we are done.