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> Let $X$ and $Y$ be topological spaces where $X$ is compact. Show that projection on the second factor $X \times Y \rightarrow Y$ is a closed map.
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> Give an example where (in the situation above) the projection to the first factor is not a closed map.
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Let $p : X \times Y \rightarrow Y$ be the projection and $C \subset X \times Y $ be closed. 

Suppose $y \not \in p(C)$.

Now $X \times \{y\} \cap C = \emptyset$, so $X \times Y − C$ is an open set containing $X \times \{y\}$, homeomorphic to $X$ and therefore compact.

For each $x \in X$ choose open sets $U\_x \times V\_x$ containing $(x, y)$ and contained in $X \times Y - C$.

The sets $U\_x, \: x \in X$ forms an open cover of the compact $X$ and so has a finite subcover $U\_{x\_1}, U\_{x\_2}, ..., U\_{x\_n}$.

If $V = \cap\_{i=1}^n V\_{x\_i}$ then $V$ is open and contains $y$.

$U\_{x\_i} \times V \subset U\_{x\_i} \times V\_{x\_i} \subset X \times Y − C 
\Rightarrow X \times V$ disjoint from $C$.

Now $V$ is disjoint from $p(C)$ and contains $y$.

We conclude $Y − p(C)$ is open and $p(C)$ is closed.

I am having trouble coming up with an example where the projection to the first factor is not a closed map. I would appreciate if anyone could point me in the right direction.