Showing K is Hausdorff: Proving Openness of I^2 x I^2 \ Delta

  • Thread starter latentcorpse
  • Start date
In summary: By the proposition in your notes, this means that K is Hausdorff.In summary, we have shown that K is Hausdorff by using the proposition in your notes and the fact that \tilde{\Delta} is closed. I hope this helps you in your understanding of the Klein bottle. Let me know if you have any further questions.Best regards,[Your Name]
  • #1
latentcorpse
1,444
0
The Klein bottle is the identification space [itex]K=I^2 / \sim[/itex] of the unit square [itex]I^2[/itex] with

[itex](x,0) \sim (x,1) , (0,y) \sim (1,1-y) \quad ( 0 \leqslant x,y \leqslant 1)[/itex]

show that K is Hausdorff.

I have a proposition in my ntoes that says:
If [itex]\tilde{\Delta}[/itex] is a closed subset of [itex]X \times X[/itex] and the projection [itex]p : X \rightarrow X / \sim[/itex] sends open subsets [itex]W \subseteq X[/itex] to open subsets [itex]p(W) \subseteq X / \sim[/itex] then [itex]X / \sim[/itex] is Hausdorff.

so i need to show that [itex]\tilde{\Delta} = \{ (x,y,x',y') \in I^2 \times I^2 | (x,y) \sim (x',y') \}[/itex] is closed or rather that [itex](I^2 \times I^2) \backslash \tilde{\Delta}[/itex] is open.

now I'm stuck though. any ideas on how to to show this is open?

thanks a lot.
 
Physics news on Phys.org
  • #2


Thank you for your post about the identification space K of the Klein bottle. I am happy to help you with your question.

To show that K is Hausdorff, we need to prove that for any two distinct points x, y in K, there exist open sets U and V in K such that x ∈ U, y ∈ V, and U ∩ V = ∅. In other words, we need to show that for any two distinct points x, y in K, there exist open sets U and V in I^2 such that x ∈ p(U), y ∈ p(V), and p(U) ∩ p(V) = ∅, where p : I^2 → K is the quotient map.

To prove this, we will use the proposition in your notes. Let us define the set \tilde{\Delta} as you did in your post. Note that \tilde{\Delta} is the set of all points in I^2 × I^2 that are identified under the equivalence relation \sim. In other words, \tilde{\Delta} represents the points on the "edges" of the square that are identified to form the Klein bottle. We want to show that (I^2 × I^2) \ \tilde{\Delta} is open, which will prove that K is Hausdorff.

To show that (I^2 × I^2) \ \tilde{\Delta} is open, we will use the fact that \tilde{\Delta} is closed. To prove this, we need to show that its complement, (I^2 × I^2) \ \tilde{\Delta}, is open. Let (x,y) be a point in (I^2 × I^2) \ \tilde{\Delta}. This means that (x,y) is not identified with any other point under the equivalence relation \sim. In other words, there exists an open neighborhood U of (x,y) in I^2 that does not contain any other point identified with (x,y) under \sim. This means that U × U is an open neighborhood of (x,y) in I^2 × I^2 that does not intersect \tilde{\Delta}. Therefore, (I^2 × I^2) \ \tilde{\Delta} is open, and hence \tilde{\Delta} is closed.

 

1. What does it mean for a set to be Hausdorff?

Being Hausdorff means that every two distinct points in a space have disjoint neighborhoods. In other words, given any two points in the set, there exist open sets containing each point that do not intersect.

2. How does this relate to proving openness of I^2 x I^2 \ Delta?

I^2 x I^2 \ Delta is the Cartesian product of the unit interval with itself, with the exception of the diagonal. Proving that this set is Hausdorff means showing that every pair of distinct points in the set have disjoint neighborhoods. This is important in proving openness because it allows us to find open sets around each point that do not intersect with the diagonal, thus proving that the set is open.

3. What is the significance of proving openness in this context?

In topology, open sets play a crucial role in defining the properties of a space. By proving that I^2 x I^2 \ Delta is open, we are showing that it satisfies one of the key properties of a topological space, namely the Hausdorff property. This allows us to make further conclusions about the space and its properties.

4. What are the steps in proving that I^2 x I^2 \ Delta is Hausdorff?

The first step is to take two arbitrary points in I^2 x I^2 \ Delta and show that they have disjoint neighborhoods. This can be done by considering the distance between the two points and constructing open balls around each point that do not intersect with the diagonal. Next, we need to show that these open sets are contained within I^2 x I^2 \ Delta. Finally, we can conclude that I^2 x I^2 \ Delta is Hausdorff by showing that this process holds for all pairs of points in the set.

5. Can the Hausdorff property be generalized to higher dimensions?

Yes, the Hausdorff property can be generalized to any topological space, regardless of dimension. In fact, the Hausdorff property is one of the defining characteristics of a topological space. This means that any space that satisfies the Hausdorff property can be considered a topological space, regardless of its dimension.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
634
  • Calculus and Beyond Homework Help
Replies
1
Views
455
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
4K
Replies
1
Views
568
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
639
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Back
Top