Is the Quotient Space X/Y Hausdorff When X is Locally Compact?

[christianrhiley]

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 \in X s.t. x and y can be separated by neighborhoods if \exists a neighborhood U of x and V of y s.t. U /\ V = \phi. Now, somehow this implies that, for open sets U, the \bigcup U \subset X are disjoint.

 

[quasar987]

Normally, a topological space is quotiented by an equivalence relation, not a subspace.

 

[mathwonk]

if you are collapsing the subspace Y to a point, maybe you want a closed subspace?

then you would be asking whether any point not in that subspace can be separated from it by disjoint open neighborhoods.

it still seems kind of odd. i guess i need more definitions. what does locally compact mean?in a hausdorff space, distinct points have disjoint open neighborhoods, so you want them to still be disjoint after collapsing Y? so you need disjoint nbhds that also miss Y? (and that's for points not themselves contained in Y...