Problem: Let X be a locally compact Hausdorff space, Y a subspace. Show that the quotient space X/Y is a Hausdorff space. My attempt at a solution: I don't have a solution. I cannot connect a Hausdorff space with a quotient space. Since X is compact Hausdorff x,y [tex]\in[/tex] X s.t. x and y can be seperated by neighborhoods if [tex]\exists[/tex] a neighborhood U of x and V of y s.t. U /\ V = [tex]\phi[/tex]. Now, somehow this implies that, for open sets U, the [tex]\bigcup[/tex] U [tex]\subset[/tex] X are disjoint.