Let $G$ be a group acting on a topological space $X$. Suppose that every $x,x' \in X$ that are not in the same orbit of the $G$-action have open neighborhood $U$ and $U'$ such tath $g(U)\cap U'=\phi$ for all $g\in G$.
$a)$. Show that $X/G$ is a Hausdorff space. $b)$. Is the quotient map $X\to X/G$ a covering map?
I just know the definitions and I have been just learning algebraic topology.
Can someone help me ?
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Same question posted here, but I too am struggling with all "hints" given: When is Quotient Map a Covering Map
Specifically, for b) it seems like we want that any points which lie in the same orbit must lie in disjoint open sets from each other. But I don't think we have that from the given condition? That is, I think we would want g(U) intersect U to be empty for all non-identity g. Or can we use the fact that the projection map is open somehow?
Some of the hints it's unclear to me whether even they are suggesting an approach to a), or a counterexample to b).....
Hint: Consider the action of the additive group of real numbers on itself via translations.