I'll extend my comment to an answer. Let $L$ be a complete lattice. Then a prime element of $L$ is an element p such that $a\wedge b\leq p$ implies $a\leq p$ or $b\leq p$. These elements are in bijection with maps from $L$ to the 2-element lattice preserving all sups and finite infs. For example, the prime elements of the lattice of ideals in a commutative ring are the prime ideals. If $A$ is a separable C*-algebra, then the prime elements of the lattice of closed 2-sided ideals are the primitive ideals (kernels of irreducible representations). If $A$ is commutative, these are the maximal ideals.
The prime elements of a lattice $L$ form a space $spec(L)$ called the spectrum of $L$. The topology has as open sets the sets D(a) with $a\in L$ where $D(a)$ consists of all prime elements $p$ with $a\nleq p$. For example, if $L$ is the lattice of ideals, this is the Zariski spectrum. If $A$ is a separable C*-algebra, then the spectrum of the closed 2-sided ideal lattice is the primitive ideal spectrum. So many of your examples are spectra of lattices.
Recall a space $X$ is sober if each irreducible closed subset has a unique generic point. Sober spaces are precisely the spectra of complete lattices. The proof $spec(L)$ is sober amounts to showing that the irreducible closed subsets are the complements of the sets D(p) with p prime and p is generic. Conversely, if X is sober, take the lattice of closed subspaces ordered by reverse inclusion. The prime elements are the irreducible closed subsets, which can be identified with points of X by taking generic points.