Product of an open set and an element of a topological group, in a transformation group, is again an open set.

Question

Suppose that $X$ is a topological space, and $T$ is a topological group which continuously acts on $X$ on the right. We call the pair $(X,T)$ a (right) transformation group. Now suppose that $\equiv$ is an “invariant” equivalence relation on $X$ in the sense that $x.t\equiv y.t$ whenever $x\equiv y$ for all $t\in T$ and for all $x,y\in X$. Now consider the quotient topology on $X/\equiv$ and let $G$ be an open set in $X/\equiv$. Does it follow that the set $\bigcup_{[x]\in G} [x.t]$ is open in $X$ for any $t\in T$?