Definition A map of topological spaces $f : X \rightarrow Y$ is called submersive if the following condition holds: $U \subset Y$ is open if and only if $f^{-1} (U)$ is open.
Let $f_1: X_1 \rightarrow Y_1$ and $f_2: X_2 \rightarrow Y_2$ be submersive maps of schemes.
Question Is $f_1 \times f_2 : X_1 \times X_2 \rightarrow Y_1 \times Y_2$ submersive?
Remark 1 This question admits topological counterpart. And I proved it (it is easy). The problem is that product of schemes (in the category of schemes) has nothing to do with their product as topological spaces.
Remark 2 I noticed that my definition (which I found in Mumford’s book on GIT ) differs from the definition from stacks project. The difference does not seem essential for my question.