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## Main Question or Discussion Point

**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.