|Feb18-11, 11:44 PM||#1|
1. The problem statement, all variables and given/known data
Prove that Hausdorff is a topological property.
2. Relevant equations
3. The attempt at a solution
For showing that a quality transfers to another space given a homeomorphism, we must show that given a Hausdorff space (X,T) and a topological space (Y,U), that (Y,U) is Hausdorff. So given two points in (Y,U), say f(x1) and f(x2), does there exist a pair of disjoint neighborhoods around both f(x1) and f(x2), call them D1 and D2. Well since (X,T) is Hausdorff, there exists a pair of disjoint neighborhoods around x1 and x2, call them O1 and O2. I am really just stuck with this problem and could use a slight push in the right direction. Any help would be greatly appreciated.
|Feb19-11, 12:23 AM||#2|
As you have a homeomrophism, you have continuous bijection between spaces, with continuous inverse
maybe try and assume the 2 disjoint sets containing x1 & x2 are mapped to a non-disjoint set and look for a contradiction
|Similar Threads for: Topological Property|
|Topological property||Calculus & Beyond Homework||1|
|HELP! Why CS is topological? Why BF is topological?||Beyond the Standard Model||1|
|Nested set property in a general topological space||Calculus & Beyond Homework||6|
|Is science a property of nature or a property of us?||General Discussion||13|