Group works on topological space

Question

I have to prove that if discrete and finite group G works on topological Hausdorff space X

$ \varphi :G \times X \rightarrow X $ and $ \varphi $ is cotinuous function, then $ X / G $ is also a Hausdorff space, where $ X / G $ is a topological space with a topology given by an orbit relation.

I only need a small hint, because it seems to be very easy, although I can't do it.Thank youfor all your answers.

Answer 1

Take two different points, $Gx$ and $Gy$ of $X/G$ where $x,y\in X$.
So these are $2$ times $|G|$ pieces of points, each distinct.
Since $X$ is Hausdorff, all these points can be separated from all the other ones by an open neighborhood.
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