> I've always wondered what it would look like [...] if the primes were mapped on to some crazy topological shape? This is easy: consider the spectrum of the

> I've always wondered what it would look like [...] if the primes were mapped on to some crazy topological shape? This is easy: consider the spectrum of the ring Z, Spec Z (by definition, the spectrum of a commutative ring consists of its prime ideals; all prime ideals of Z (with the exception of the zero ideal) are generated by the prime numbers; in this sense, we can identify all elements of Spec Z (with the exception of the zero ideal, which is "just another point" of Spec Z) with the prime numbers if we desire). On Spec Z, we can define a topology, the Zariski topology, by defining V(I) := {P \in Spec Z | I \subseteq P}, where I is an ideal of Z, as the closed sets of the Zariski topology. See * https://en.wikipedia.org/w/index.php?title=Zariski_topology&oldid=1091588561#Spectrum_of_a_ring https://en.wikipedia.org/w/index.php?title=Zariski_topology&... * https://en.wikipedia.org/w/index.php?title=Spectrum_of_a_ring&oldid=1076963164#Zariski_topology https://en.wikipedia.org/w/index.php?title=Spectrum_of_a_rin... for details.