 Positive orbit separation implies a Hausdorff quotient

Suppose a group acts by isometries on a metric space $X$, and for points $a,b$ in different orbits one has ${\inf }_{g}d(a,gb)>0$. Then the orbit space with its quotient topology is Hausdorff. Indeed, choose $\epsilon$ less than one third of this positive infimum and saturate the open radius-$\epsilon$ balls about $a$ and $b$ by the action. These saturated sets are open because every acting isometry is a homeomorphism. Their translates cannot meet: their centres stay at least the given positive distance apart. Their disjoint open images in the quotient separate the two orbit points, since the preimage of each image is its saturated set. This applies to the glide reflection $(x,y)\mapsto (x+1,-y)$ on a closed strip: the distance from one point to the translates of another tends to infinity as the integer translation grows, so two distinct orbits have a positive minimum distance.