Lemma 1

If a topological group $G$ operates continuously on a topological space $X$, then for each $s \in G$ the mapping $x \to s.x$ is a homeomorphism of $X$ onto $X$.

For this mapping is a continuous bijection whose inverse $x \to s^{-1}.x$ is also continuous.

We recall that for each $x \in X$ the set $G.x$ of transforms $s.x$ of $x$ by the elements $s$ of $G$ is called the orbit of $x$ (with respect to the group of operators $G$), and that the set of all $s \in G$ such that $s.x = x$ is a subgroup of $G$ called the stabilizer of $x$. The relation $R(x, y) : "y$ belongs to the orbit of $x"$ is an equivalence relation on $X$, called the equivalence relation defined by $G$; the equivalence classes with respect to this relation are the orbits of the points of $X$. The topological space $X/R$ is called the orbit space of $X$ (with respect to $G$), or the quotient space of $X$ by the group $G$, and is denoted by $X/G$; and the topology of $X/G$ is said to be the quotient of the topology of $X$ by $G$.