regions naturally arise in the context of fundamental sets of solutions of linear ordinary differential equations depending on a parameter. Moreover, several properties of such equations depend on whether the underlying domain $\Omega$ of the independent variable $x$ and the parameter $t$ is $x$-simple and how the $x$-simplicity is combined with the connectedness, in particular, whether all the connected components of $\Omega$ or at least some of them are connected or whether the domain $\Omega$ has $x$-simple pieces. These properties include

-   the existence of fundamental sets of solutions and of sets of solutions with nonvanishing Wronskians,

-   the relation between these two kinds of solution sets,

-   the existence of solutions that are not identically zero for such homogeneous equations and

-   the general existence of solutions for such inhomogeneous equations.

See Figure \[fig:VariationsOnXSimplicityAndConnectedness\] for some variants of combining $x$-simplicity, connectedness and simple connectedness.

![Variants of combining $x$-simplicity, connectedness and simple connectedness: ([*a*]{}) connected, $x$-simple and thus simply connected set; ([*b*]{}) connected, non-$x$-simple and thus multiply connected set; ([*c*]{}) non-$x$-simple, simply connected set; ([*d*]{}) disconnected $x$-simple set (each of its connected components is necessarily $x$-simple and thus simply connected); ([*e*]{}) disconnected non-$x$-simple set whose connected components are $x$-simple and 