Hausdorffness -quotient space of a hausdorff space

by Tomer
Tags: hausdorff, hausdorffness, quotient, space
Tomer is offline
Jan15-12, 04:03 PM
P: 202
1. The problem statement, all variables and given/known data

Let (X,[itex]\tau[/itex]) be a hausdorff space and let ~ be an equivalence relation, so that all equivalence classes are finite. Prove: X/~ is also a hausdorff space.

2. Relevant equations
Here's the definition of the standard topology of X/~:
Let q:X-> X/~ be the function that sends x to [x] (its equivalence class). U is called "open" in X/~ if q-1(U) is open in X.

3. The attempt at a solution

Sounds like a simple enough exercise , but unfortunately I can't solve it.
I started out of course with taking two distinct points, [x], [y] in X/~.
We know that [x], [y] are finite.
Now, at this point I'll assume [x] = {x} and [y] = {y}. It's not a necessary simplification, it's just going to save some messy writing... otherwise I'd have to deal with lots of sets and indexes. If I can solve it for this example I should be able to solve the general case (that [x] and [y] contain n and m elements each).
My goal is to find two open distinct sets around [x] and [y] in X/~.
We know that since [x] ≠ [y], x ≠ y. Therefore, we have two distinct open sets U, V in X around x and y respectively.
That's the part where I'm not sure anymore what to do.
I tried taking the obvious parallel sets in X/~ according to this definition:
For a set U in X define: [U] = {[x] : [itex]x \in U[/itex]}.
So I would expect that [U] and [V] would be the two desired sets in X/~. Unfortunately, I cannot prove that they are distinct, nor do I think it's necessarily correct.

Does anyone have an idea here?

Thanks a lot!

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