# Homework Help: Hausdorff Spaces

1. Apr 3, 2010

### latentcorpse

The Klein bottle is the identification space $K=I^2 / \sim$ of the unit square $I^2$ with

$(x,0) \sim (x,1) , (0,y) \sim (1,1-y) \quad ( 0 \leqslant x,y \leqslant 1)$

show that K is Hausdorff.

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

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

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

thanks a lot.