I've been studying the Nullstellensatz and the Zariski topology.
I understand the basic gist is that in $\Bbb C[x_1, x_2, ..., x_n]$, varieties are in one-to-one correspondence with radical ideals. Each of these is in turn in a one-to-one correspondence with the set of prime ideals supersetting it. So, the (original) Zariski topology directly on $\Bbb C^n$ translates to the (modern) Zariski topology on $\text{Spec}(\Bbb C[x_1, x_2, ..., x_n])$.
I also get that the next step is to look at the topology on $\text{Spec}(R)$ directly for some arbitrary commutative ring $R$. The "varieties" of the ring are again given by the radical ideals, the (non-generic) "points" are the maximal ideals, etc.
Schemes, then, are often described as representing varieties "with multiplicity," so that in $\Bbb C[x]$, $((x-1)^2)$ generates a different scheme than $(x-1)$. But, my perspective has been that varieties map to radical ideals map to sets of prime ideals in the spectrum. I can get that schemes are arbitrary ideals, rather than radical ideals, but what do they map to in the spectrum? The set of prime ideals containing $((x-1)^2)$ is exactly the same as $(x-1)$, so how are these represented as different schemes in the spectrum?
Basically: how are schemes represented in the spectrum? Are they also closed under the Zariski topology?