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**Gamma** c) $\displaystyle \Delta = \{(x,x)\in \mathbb{R}^2|x\in \mathbb{R}\}$ This is the diagonal and it is probably a theorem you have done that is pretty easy to show from definitions that hausdorff iff diagonal is closed so $\displaystyle \Delta$ is closed in $\displaystyle \mathbb{R}^2$ but its projection is $\displaystyle \mathbb{R}$ which is open.

If you dont believe me that $\displaystyle \Delta$ is closed, just look at $\displaystyle \mathbb{R}^2-\Delta$ and see why it's open.