I want to show $p_x: X\times\ Y \to X$ is an open map. Here's my proof:
Let $W \subset\ X\times\ Y$ be open subset, then $W = \bigcup U_\alpha \times\ V_\beta$, for $U_\alpha, V_\beta$ are open subsets of $X, Y$ respectively.
Then $p_x(W) = p_x (\bigcup U_\alpha \times\ V_\beta)= \bigcup p_x (U_\alpha \times\ V_\beta) = \bigcup U_\alpha$ is open in $X$, so $p_x$ is an open map/
Yes, your proof perfectly works. Here is a related question, if you want to see.
Notice that the projections are not closed in general. (For instance, the graph $G$ of $f:x\mapsto 1/x$ ($f$ being defined on $\Bbb R \setminus\{0\}$) is closed in $\Bbb R^2$ endowed with the usual topology, whereas the projection of $G$ on the $x$-axis is open, because it is $\Bbb R \setminus\{0\}$. However, Kuratowski-Mrówka theorem (see for instance here) states that the projection $p : X \times Y \longrightarrow Y$ is closed for all topological spaces $Y$ iff $X$ is compact).
