A properly discontinuous action on a space $Y$.

Question

I am unsure as to the meaning of the following notation.

Suppose $H$ acts properly discontinuously on a space $Y$.

Then the space $H \setminus Y =Y \setminus (hy \sim y)$ $\dots$ with the quotient topology

How do I interpret these two spaces?

The LHS is $Y$ modded out by the group $H$ which makes sense. On the RHS, are we modding $Y$ by $ \{ hy \sim y : \forall h \in H, \hspace{2mm} \forall y \in Y \}$ ?

Answer 1

I think the LHS is just notation for the RHS, so there's only one space.

You can define a relation $\sim$ on $Y$ by saying $a \sim b$ if and only if $b = ha$ for some $h \in H$. You can check that this is an equivalence relation.

Then you define a topology on the set of equivalence classes of $\sim$ by the quotient topology.