1. ## Hausdorff space

Show that X is a Hausdorff space iff. the diagonal $\displaystyle \Delta=\{x\times x:x\in X\}$ is a closed set in the product space.

$\displaystyle \Rightarrow$
Suppose X is a Hausdorff space. Then $\displaystyle \forall x_1,x_2\in X$ such that $\displaystyle x_1\neq x_2$ there exists neighborhoods $\displaystyle U_1 \ \text{and} \ U_2$ of $\displaystyle x_1, x_2$, respectively, such that $\displaystyle U_1\cap U_2=\O$.

I don't know how I am supposed to use this to conclude the other piece.

2. ## Re: Hausdorff space

Originally Posted by dwsmith
Show that X is a Hausdorff space iff. the diagonal $\displaystyle \Delta=\{x\times x:x\in X\}$ is a closed set in the product space.
If $\displaystyle (x,y)$ is such that $\displaystyle x\ne y$ can you show that there is a ball centered at $\displaystyle (x,y)$ that contains no point of $\displaystyle \Delta$.
Hint: $\displaystyle \frac{|x-y|}{4}>0.$

3. ## Re: Hausdorff space

Originally Posted by Plato
If $\displaystyle (x,y)$ is such that $\displaystyle x\ne y$ can you show that there is a ball centered at $\displaystyle (x,y)$ that contains no point of $\displaystyle \Delta$.
Hint: $\displaystyle \frac{|x-y|}{4}>0.$
I don't think I can use that. I don't know it is a Metric space.

4. ## Re: Hausdorff space

Originally Posted by dwsmith
Show that X is a Hausdorff space iff. the diagonal $\displaystyle \Delta=\{x\times x:x\in X\}$ is a closed set in the product space.

$\displaystyle \Rightarrow$
Suppose X is a Hausdorff space. Then $\displaystyle \forall x_1,x_2\in X$ such that $\displaystyle x_1\neq x_2$ there exists neighborhoods $\displaystyle U_1 \ \text{and} \ U_2$ of $\displaystyle x_1, x_2$, respectively, such that $\displaystyle U_1\cap U_2=\O$.

I don't know how I am supposed to use this to conclude the other piece.
Merely note that if $\displaystyle \Delta_X$ is closed, then $\displaystyle X^2-\Delta_X$ is open, and so if $\displaystyle x\ne y\in X$ then $\displaystyle (x,y)\in X^2-\DeltaX$ and so there exists some basic open neighbohood $\displaystyle U\times V$ of $\displaystyle (x,y)$ with $\displaystyle U\times V\subseteq X^2-\Delta_X$. Conclude from this that $\displaystyle x\in U$, $\displaystyle y\in V$, and $\displaystyle U\cap V=\varnothing$.

Now, try to reverse this to get the reverse implication you asked about.