Hausdorffness quotient space of a hausdorff spaceby Tomer Tags: hausdorff, hausdorffness, quotient, space 

#1
Jan1512, 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! Tomer. 


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