Let $X$ be a set, let $G$ be a group and let $\alpha: G \times X \rightarrow X$ be a left group action of G on $X$. Now let $X/G$ be the the set of all orbits of $X$ under the action of $G$. Then if $X$ is a space, how is the topology of $X/G$ defined?
Source. I am trying to understand the definition of a Lens space, which is defined as $S^3/\mathbb{Z}_p$. But Wikipedia only defines the set, not the topology.
Also is the topology of $X/G$ defined for all groups $G$, or some special class?
The standard way is this: if $\pi\colon X\longrightarrow X/G$ is the map that maps each $x\in X$ into its orbit, then the topology that we consider on $X/G$ is the final topology with respect to $\pi$. In other words, it's the topology$$\{A\subset X/G\mid\pi^{-1}(A)\text{ is an open subset of }X\}.$$