Prove Hausdorff is a Topological Property

In summary, to prove that Hausdorff is a topological property, we must show that given a Hausdorff space (X,T) and a topological space (Y,U), (Y,U) is also Hausdorff. This can be done by showing that given two points in (Y,U), there exist two disjoint neighborhoods around them. As a homeomorphism implies a continuous bijection between spaces, we can assume that the two disjoint sets containing the points in (Y,U) are mapped to a non-disjoint set, and then look for a contradiction. Any further assistance would be appreciated.
  • #1
tylerc1991
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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.
 
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  • #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
 

FAQ: Prove Hausdorff is a Topological Property

1. What is the Hausdorff property?

The Hausdorff property is a topological property that states that for any two distinct points in a topological space, there exist disjoint open sets containing each point. In other words, the space is "well-separated" and any two points can be separated by open sets.

2. Why is the Hausdorff property important in topology?

The Hausdorff property is important because it allows for a more precise understanding and classification of topological spaces. It also has many useful applications in other areas of mathematics, such as algebraic geometry and functional analysis.

3. How can one prove that a space has the Hausdorff property?

To prove that a space has the Hausdorff property, one must show that for any two distinct points, there exist disjoint open sets containing each point. This can be done by constructing such open sets or by showing that the space satisfies the definition of the Hausdorff property.

4. Can a space have the Hausdorff property and not be a topological space?

No, the Hausdorff property is a topological property and is only defined for topological spaces. A space cannot have the Hausdorff property without also being a topological space.

5. Are all metric spaces Hausdorff?

Yes, all metric spaces are Hausdorff. In fact, the Hausdorff property is a stronger condition than metric spaces, as it can be defined for more general topological spaces.

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