Prove Hausdorff is a Topological Property

[tylerc1991]

Homework Statement

Prove that Hausdorff is a topological property.

Homework Equations

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.

 

[lanedance]

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