  We study Zariski-like topologies on a proper class $X\varsubsetneqq L$ of a
complete lattice $\mathcal{L}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the
so called classical Zariski topology $(X,\tau ^{cl})$ and study its topological
properties (e.g. the separation axioms, the connectedness, the compactness) and
provide sufficient conditions for it to be $\textit{spectral}$. We say that
$\mathcal{L}$ is $X$\emph{-top} iff% \begin{equation*} \tau :=\{X\backslash
V(a)\mid a\in L\},\text{ where }V(a)=\{x\in L\mid a\leq x\} \end{equation*}% is
a topology. We study the interplay between the \textit{algebraic properties} of
an $X$-top complete lattice $\mathcal{L}$ and the $\textit{% topological
properties}$ of $(X,\tau ^{cl})=(X,\tau ).$ Our results are applied to several
spectra which are proper classes of $\mathcal{L}% :=LAT(_{R}M)$ where $M$ is a
left module over an arbitrary associative ring $% R$ (e.g. the spectra of
prime, coprime, fully prime submodules) of $M$ as well as to several spectra of
the dual complete lattice $\mathcal{L}^{0}$ (e.g. the spectra of first, second
and fully coprime submodules of $M$).
