My question seems far too basic to be unknown, but I could not find anything relevant...
Let $X$, $Y$ and $Z$ be compact connected metric spaces, and let $F \subset X \times Y$ and $G \subset Y \times Z$ be closed binary relations such that both projections are surjective. Suppose that they are also connected as topological spaces.
Is it true that $G \circ F$ is connected?
Probably I should also mention that a combinatorial version of this statement for connected graphs is known to be true.
No. There are two (discontinuous) surjective maps $f,g:S^1\to S^1$ whose graphs are connected but the graph of $g\circ f$ (as well as its closure) is not.
The map $f$ is defined as follows, using the standard parametrization of $S^1$ by $\mathbb R/2\pi\mathbb Z$. It is the identity on the complement from the arc (parametrized by) $[\pi/2,\pi]$. The arc $[\pi/2,\pi]$ is slightly stretched from the point $\pi/2$ so that its image is the arc $[\pi/2,\pi+\varepsilon]$. The map $g$ is similar but with a discontinuity at 0 rather than $\pi$.
The graphs of $f$ and $g$ can be parametrized by an interval and hence connected. The graph of $g\circ f$ is not, because the map has two simple discontinuities (at 0 and $\pi$) that divide the circle into two components.
The graphs are not closed in $S^1\times S^1$, but adding points $(\pi,\pi)$ and $(0,0)$ fixes this issue.
