Hausdorf space condition problem

[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

Homework Equations

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?

 

[fresh_42]

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.

 

[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?

 

[fresh_42]

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.

 

[fresh_42]

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.

 

[WWGD]

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]

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?

 

[fresh_42]

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.

 

[WWGD]

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).

 

[PsychonautQQ]

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 ?

 

[WWGD]

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.