the equation $\mathcal P$: $Pu=0$ belong to ${\rm C}^p_x(\Omega)$.

Expressions of the form $\zeta\psi$ for some functions $\zeta\in {\rm C}({\mathop{\rm pr}\nolimits}_t\Omega)$ and $\psi\in {\rm C}(\Omega)$ should always be interpreted as the product of $\psi$ by the pullback of $\zeta$ to $\Omega$ with respect to the map ${\mathop{\rm pr}\nolimits}_t\big|_\Omega$.

As already noted above, for a function $u$ of $(t,x)$ we set $u_i:={\partial}^i u/{\partial}x^i$, $i\in\mathbb N$, $u_0:=u$, and ${\partial}_x={\partial}/{\partial}x$. We also employ, depending upon convenience or necessity, the notation $u_x=u_1$. By $\mathrm W(\varphi^1,\dots,\varphi^p)$ we denote the Wronskian of functions $\varphi^1,\dots,\varphi^p\in {\rm C}^p_x(\Omega)$ in the variable $x$, i.e., $\mathrm W(\varphi^1,\dots,\varphi^p)=\det({\partial}_x^{s'-1}\varphi^s)_{s,s'=1,\dots,p}$.

The indices $s$, $s'$ and $s''$ run from 1 to $p$, and summation with respect to repeated indices is always understood.

The plan of the paper is as follows: In Section \[sec:xSimpleSets\] we introduce sets simple with respect to a variable (in our case, $x$), and prove some basic topological properties of such sets. In Section \[sec:FundamentalSolutionSetsOfLinODEsWithParameter\] we provide an appropriate notion of fundamental sets of solutions to homogeneous linear parameter-dependent ODEs and relate it to the nonvanishing of the corresponding Wronskian. We also characterize both concepts in terms of $x$-simplicity of (pieces of) the underlying domain $\Omega$. The inhomogeneous setting is studied 