Does the poset of all partitions of the real numbers (ordered by fineness), $\text{Prt}\,\mathbf{R}$, contain a copy of $\mathbf{R}$? I'm imagining something analogous to the discrete topology on $\mathbf{R}$. More specifically, I'm thinking about a terminal object in the canonical category induced by the posetal structure on $\text{Prt}\,\mathbf{R}$.
If we have a correspondence between partitions of $\mathbf{R}$ and surjections $f \colon \mathbf{R} \twoheadrightarrow A$, then the identity function seems a perfectly good candidate for a partition: $\text{id}_\mathbf{R} \colon \mathbf{R} \twoheadrightarrow \mathbf{R}$.
I'd appreciate anyone dropping in the standard terminology or directions to appropriate literature.
There's nothing special about the reals here, so I'll just talk about an arbitrary set $X$.
Basically, yes.
The finest partition on a set $X$ isn't exactly $X$ itself; rather, it's $$\{\{x\}: x\in X\}$$ ($\{x\}$ and $x$ are distinct objects). If we choose to represent partitions by functions, then this partition is indeed represented by the identity map on $X$.
As far as terminology goes, I don't think there is a standard one - see the discussion here for example - and in particular "trivial partition" is used to refer to both this partition and the partition with a single class. In my opinion the phrase "discrete partition" is unambiguous, but I don't think it's universally used (although I have seen it used sometimes).