Let $G$ be a compact topological group, and $X$ be a Hausdorff space. We assume that $G$ acts on $X$. Is the quotient space $X/G$ with the quotient topology a Hausdorff space? It seems that the answer is YES from this post How can I prove formally that the projective plane is a Hausdorff space? , if we assume further that $X$ is locally compact. My question is whether this locally compactness condition can be removed.