# Proving the Hausdorff Property: The Diagonal in X x X

• Symmetryholic
In summary, the diagonal of a space X is closed if and only if the complement of X\setminus\Delta is open.
Symmetryholic

## Homework Statement

Show that X is Hausdorff if and only if the diagonal $$\Delta = \{x \times x | x \in X \}$$ is closed in $$X \times X$$.

## Homework Equations

Definition of Hausdorff Space (T2) : A topological space in which distinct points have disjoint neighborhoods.

## The Attempt at a Solution

Symmetryholic said:

## Homework Statement

Show that X is Hausdorff if and only if the diagonal $$\Delta = \{x \times x | x \in X \}$$ is closed in $$X \times X$$.

## Homework Equations

Definition of Hausdorff Space (T2) : A topological space in which distinct points have disjoint neighborhoods.

## The Attempt at a Solution

With so little work on your part shown to go by it's difficult to know where you're stuck.

If you haven't already, express the closedness of $$\Delta$$ in $$X\times X$$ (which I assume has the product topology) in terms of the openness of its complement.

Now have a go at proving each direction (neither is more difficult than the other) if you haven't already, and please show us your efforts.

Unco said:
If you haven't already, express the closedness of $$\Delta$$ in $$X\times X$$ (which I assume has the product topology) in terms of the openness of its complement.

->
If X is Hausdorff, the diagonal $$\Delta$$ is closed in $$X\times X$$.

Assume X is Hausdorff. Now, we have two distinct points x, y and disjoint open sets U, V containing x, y, respectively. The basis element $$U \times V$$ containing $$(x,y) \in X \times X$$ should not intersect $$\Delta$$ by the assumption given to the Hausdorff property.
For every $$(x,y) \notin \Delta$$, we have a basis element in $$X \times X$$ containing (x,y), which does not intersect $$\Delta$$.
Thus, $$X \times X \setminus \Delta is open$$ and we conclude $$\Delta$$ is a closed set in $$X \times X$$ .

<-
If the diagonal $$\Delta$$ is closed in $$X\times X$$, X is Haudorff.

Supppose $$\Delta$$ is closed in $$X\times X$$. Then, $$X \times X \setminus \Delta$$ is open. Let $$(x,y) \in X \times X$$ and $$x \neq y$$. For $$(x,y) \notin \Delta$$, we have a basis element $$U \times V$$ in $$X \times X$$ containing (x, y).
We remain to show U and V are disjoint. Suppose on the contrary that U and V are not disjoint. Then, there is an element $$(z,z) \in X times X$$ which belongs to both U and V. Contradicting the fact that x and y are distinct.
Thus, X is Hausdorff.

Symmetryholic said:
->
If X is Hausdorff, the diagonal $$\Delta$$ is closed in $$X\times X$$.

Assume X is Hausdorff. Now, we have two distinct points x, y and disjoint open sets U, V containing x, y, respectively. The basis element $$U \times V$$ containing $$(x,y) \in X \times X$$ should not intersect $$\Delta$$ by the assumption given to the Hausdorff property.
For every $$(x,y) \notin \Delta$$, we have a basis element in $$X \times X$$ containing (x,y), which does not intersect $$\Delta$$.
Thus, $$X \times X \setminus \Delta is open$$ and we conclude $$\Delta$$ is a closed set in $$X \times X$$ .

<-
If the diagonal $$\Delta$$ is closed in $$X\times X$$, X is Haudorff.

Supppose $$\Delta$$ is closed in $$X\times X$$. Then, $$X \times X \setminus \Delta$$ is open. Let $$(x,y) \in X \times X$$ and $$x \neq y$$. For $$(x,y) \notin \Delta$$, we have a basis element $$U \times V$$ in $$X \times X$$ containing (x, y).
We remain to show U and V are disjoint. Suppose on the contrary that U and V are not disjoint. Then, there is an element $$(z,z) \in X times X$$ which belongs to both U and V. Contradicting the fact that x and y are distinct.
Thus, X is Hausdorff.
That's quite a leap from your previous post.

Now how 'bout you tackle your https://www.physicsforums.com/showthread.php?t=287038" without copying down the solution from an external source. It's the only way you'll learn topology.

Last edited by a moderator:
Can you please give me a link of an external source you mentioned?
It's my self study of topology (I am not even majoring in math) and I don't need to copy the external source to show you something to impress you. Rather, I just did for my self study purpose and asked an advice if someone finds an error in my attempt to the solution.

Symmetryholic said:
Can you please give me a link of an external source you mentioned?
It's my self study of topology (I am not even majoring in math) and I don't need to copy the external source to show you something to impress you. Rather, I just did for my self study purpose and asked an advice if someone finds an error in my attempt to the solution.
Well, in that case, your work is quite error-free indeed! I would only suggest rephrasing picking an element $$(x,y)\not\in \Delta$$ as picking an element $$(x,y)\in (X\times X)\backslash \Delta$$. Now, on to your other problem!

## 1. What is the Hausdorff property?

The Hausdorff property, also known as the Hausdorff axiom or the separation axiom, is a fundamental concept in topology. It states that for any two distinct points in a topological space, there exists open sets containing each point that do not overlap.

## 2. What is the diagonal in X x X?

The diagonal in X x X is a subset of the Cartesian product of a topological space X with itself. It consists of all pairs (x, x) where x is an element of X. In other words, it is the set of all points in X x X that have the same value for both coordinates.

## 3. Why is proving the Hausdorff property important?

The Hausdorff property is important because it guarantees that a topological space has enough separation between points to allow for the development of important mathematical concepts and theorems. It also allows for the construction of more complex topological spaces by using the Hausdorff property as a building block.

## 4. What is the significance of the diagonal in proving the Hausdorff property?

The diagonal plays a crucial role in proving the Hausdorff property because it allows us to show that any two distinct points in a topological space can be separated by open sets. By using the diagonal, we can construct open sets that contain one point and not the other, fulfilling the requirements of the Hausdorff property.

## 5. How is the Hausdorff property proven using the diagonal in X x X?

The proof of the Hausdorff property using the diagonal in X x X involves using the fact that the diagonal is a closed set in X x X. This means that its complement, which consists of all points that are not on the diagonal, is an open set. By showing that this complement contains the open sets needed to separate any two distinct points, we can prove that the Hausdorff property holds for the topological space X.

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