# Hausdorf space condition problem

• PsychonautQQ
In summary: These maps are surjective, so given any two points in the space, there is a unique open set containing both of them.
PsychonautQQ

## Homework Statement

Show that X is a Hausdorff space IFF the 'diagnol of x' given by t = {(x,x) | X * X} is closed as a subspace of X*X

## The Attempt at a Solution

So since X Is Hausdorff so is X*X and t, because the product of two Hausdorff spaces if Hausdoff and the subspace of any Hausdorff space is Hausdorf.

So I've been doing a lot of thinking about this and have some ideas, so here we go.

Since X is Hausdorff, for any x_1 and x_2 in X, there exists open sets (or neighborhoods) U_1 and U_2 that contain x_1 and x_2 respectively and their intersection is the empty set.

To show that t is closed as a subset of X*X, we must show that t contains all it's limit points. It is perhaps worth noting that if A is a subset of a Hausdorff space Y, then b is a limit point of A iff every neighborhood of b intersects A at infinitely many points.

First I will try to show that X being a Hausdorff space implies that t is closed:

I will proceed by contradiction
Suppse X is a Hausdorff space and that t is not closed. Then there exists an element in the closure of t (hereby known as ct) that is not in t, we will call this element y=(x_1,x_2) where x_1 does not equal x_2, for if it did then y would be in ct. Since y is in the closure of t, every neighborhood of y contains some element of t, or said otherwise, if U is a neighborhood of y, then the intersection of U and t is nonempty.

Question: y must be of the form (x_1,x_2), correct? I am confused by that notion because that would mean that y is an element of X*X, which is not necesarily true I don't think, but rather it is only gaurenteed to be on the closure of X*X.

If y must be of the form (x_1,x_2) where each x_i is an element of X then I smell a contradiction lingering closely...

Anyone have any thoughts/concerns?

Life would be easier if you forgot about the limit points and prove ##X \text{ Hausdorff } \Longleftrightarrow (X \times X) - t \text{ open }##. Points ##(x_1,x_2)## outside ##t## automatically fulfill ##x_1 \neq x_2## and open neighborhoods occur in the definition of open sets as well as of Hausdorff spaces.

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PsychonautQQ
X is Hausdorff so for any points in X, say x_1 and x_2, there exists neighborhoods U_1 and U_2 such that x_1 is in U_1 and x_2 is in U_2 and U_1 and U_2 are disjoint.

(X * X) - t must be open because for any point (x_1,x_2) in this space there exists an open neighborhood U_1 (Union) U_2 the contains the point.

Am I close?

PsychonautQQ said:
X is Hausdorff so for any points in X, say x_1 and x_2, there exists neighborhoods U_1 and U_2 such that x_1 is in U_1 and x_2 is in U_2 and U_1 and U_2 are disjoint.

(X * X) - t must be open because for any point (x_1,x_2) in this space there exists an open neighborhood U_1 (Union) U_2 the contains the point.

Am I close?
The track is correct, but "union" false. How do open sets in ##X \times X## look like? And how can you use ##U_1\, , \,U_2## here?

PsychonautQQ
Open sets in X*X are tuples that look like (x_i,x_j) where each x is an element in X. perhaps (U_1,U_2) is an open neighborhood that contains the point (x_1,x_2), or I think the correct notation would actually be U_1 * U_2.

PsychonautQQ said:
Open sets in X*X are tuples that look like (x_i,x_j) where each x is an element in X. perhaps (U_1,U_2) is an open neighborhood that contains the point (x_1,x_2), or I think the correct notation would actually be U_1 * U_2.
Yes, that's the key object for both directions. Correct would have been ##U_1 \times U_2 \subseteq X \times X## as an open set, not U_1 * U_2. One doesn't write the product space with a dot, at least I haven't ever seen it. Now why is ##U_1 \times U_2 \subseteq X \times X - t## if ##X## is Hausdorff? This is needed as we want to find an open neighborhood of ##(x_1,x_2)## that is completely in our presumably open set ##X \times X - t ##.

The other direction is similar.

PsychonautQQ
PsychonautQQ said:
Open sets in X*X are tuples that look like (x_i,x_j) where each x is an element in X. perhaps (U_1,U_2) is an open neighborhood that contains the point (x_1,x_2), or I think the correct notation would actually be U_1 * U_2.
Think of the projection maps ##\pi_1, \pi_2 ; \pi_1(x,y)=x , \pi_2(x,y)=y ##.

PsychonautQQ
fresh_42 said:
Yes, that's the key object for both directions. Correct would have been ##U_1 \times U_2 \subseteq X \times X## as an open set, not U_1 * U_2. One doesn't write the product space with a dot, at least I haven't ever seen it. Now why is ##U_1 \times U_2 \subseteq X \times X - t## if ##X## is Hausdorff? This is needed as we want to find an open neighborhood of ##(x_1,x_2)## that is completely in our presumably open set ##X \times X - t ##.

The other direction is similar.

U_1 x U_2 is Hausdorff because it is the subspace of a Hausdorff space, is that what you are asking?

PsychonautQQ said:
U_1 x U_2 is Hausdorff because it is the subspace of a Hausdorff space, is that what you are asking?
No, I meant an open set in ##X \times X## looks like ##U_1 \times U_2## with open sets ##U_i \subseteq X## by the definition of the product topology. Now all you need is a connection that relates ##x_1 \in U_1## and ##x_2 \in U_2## as the open sets necessary to establish the Hausdorff property in one direction, and the open property of ##X \times X - t## necessary for the other direction. ##x_1 \neq x_2## resp. the fact that ##U_1 \cap U_2 = \emptyset## has to be somehow related to this diagonal, which by the way would have been better noted as ##\Delta## rather than by ##t##, which could be overlooked too easily, especially if you don't use LaTeX to write it.

Maybe you should restart from scratch:

Hausdorff ##\Longleftrightarrow \; \forall \;x_1\neq x_2 \;\exists\; U_1(x_1)\; , \;U_2(x_2) \text{ open }\, : \,U_1 \cap U_2 = \emptyset##

1. Find an open set of ##U(x_1,x_2) \subseteq X \times X- \Delta## that doesn't contain any points of ##\Delta =t##.
2. Given ##(x_1,x_2) \in X \times X - \Delta ## and ##X \times X - \Delta## is open, show there are open sets ##U_i(x_i) \subseteq X## with ##U_1(x_1) \cap U_2(x_2) = \emptyset##.
The entire proof is to see, how the diagonal plays a role in here.

PsychonautQQ
PsychonautQQ said:
U_1 x U_2 is Hausdorff because it is the subspace of a Hausdorff space, is that what you are asking?
Think of this, if ##U_x ## and ##U_y ## are disjoint. Can there be an element ##(a,a)## in ## U_x \times U_y ## ? If this is the case then a is in...( Use projection maps here).

Wow! you guys are such good explainers, thanks so much! I really need to learn LaTeX...

So showing that every element in X*X - t, so a generic element of the form (x_i,x_k) where i does not equal k, is contained in an open neighborhood U_i*U_k that is completely contained in X*X - t will show that X*X - t is open in X*X ?

PsychonautQQ said:
Wow! you guys are such good explainers, thanks so much! I really need to learn LaTeX...

So showing that every element in X*X - t, so a generic element of the form (x_i,x_k) where i does not equal k, is contained in an open neighborhood U_i*U_k that is completely contained in X*X - t will show that X*X - t is open in X*X ?

Yes; if you can show that a generic point is contained in an open set S where S is contained entirely in the complement, this is saying the complement is open and therefore the diagonal is closed. But note that S can be of any form as long as it is open.

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## 1. What is the Hausdorff space condition problem?

The Hausdorff space condition problem is a fundamental question in topology and set theory, which asks whether every topological space can be embedded into a Hausdorff space in a unique way. In simpler terms, it is a question about the existence and uniqueness of the Hausdorffization of a given topological space.

## 2. Why is the Hausdorff space condition problem important?

The solution to the Hausdorff space condition problem has significant implications in many areas of mathematics, including topology, set theory, and algebraic geometry. It is also relevant in computer science and physics, as it helps to understand the structure and properties of spaces and their representations.

## 3. What is the current status of the Hausdorff space condition problem?

The Hausdorff space condition problem remains an open problem, with no definitive solution yet. However, there have been significant developments and progress made towards understanding and solving the problem, including the study of topological invariants and the use of different set-theoretic axioms.

## 4. What are some proposed solutions to the Hausdorff space condition problem?

Some proposed solutions to the Hausdorff space condition problem include the use of forcing techniques in set theory, the development of new topological concepts such as quasi-Hausdorff spaces, and the application of algebraic topology methods. However, none of these solutions have been proven to be a complete answer to the problem.

## 5. How does the solution to the Hausdorff space condition problem impact other areas of mathematics?

The solution to the Hausdorff space condition problem would have significant implications for other areas of mathematics, including algebraic geometry, differential geometry, and functional analysis. It would also provide a better understanding of topological spaces and their properties, which can be applied in various fields of science and engineering.

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