Remark 1

The projection of a closed subset of $X_1 \times X_2$ onto a factor need not be closed. For example, in the rational plane $\mathbf{Q}^2$, the hyperbola whose equation is $x_1 x_2 = 1$ is a closed set, but both its projections are equal to the complement of the point 0 in $\mathbf{Q}$, and this is not a closed set.

(*) By the section $A(x_1)$ of $A$ at $x_1$ is meant the set of all $x_2 \in X_2$ such that $(x_1, x_2) \in A$ (cf. Set Theory R, \S\ 3, no. 7).