I'm reading Deitmar's paper on Schemes over $\mathbb{F}_1$. Proposition 2.4. states that for a scheme $X$ over $\mathbb{F}_1$ there is a bijection between $X(\mathbb{F}_1)$ and the set of connected components of $X$. I don't understand the proof, which is quite sketchy. Here is what I think:

Elements of $X(\mathbb{F}_1)$ correspond to morphisms $\mathrm{Spec}(\mathbb{F}_1) \to X$, where $\mathrm{Spec}(F_1)$ is the point together with the trivial monoid sheaf $1$. These morphisms correspond to a point $x \in X$ together with a local homomorphism $\mathcal{O}_{X,x} \to \{1\}$. But this is unique and exists iff $\mathcal{O}_{X,x} = \mathcal{O}_{X,x}^*$, i.e. iff the stalk is actually a group. Now to such a point we should associate to irreducible closed subset $\overline{\{x\}} \subseteq X$. But why should it be a connected component, and why does every one arise this way?

I can show that every irreducible scheme over $\mathbb{F}_1$ has exactly one generic point. So perhaps Proposition 2.4 should talk about irreducible components? I'm a bit confused. Also Deitmar's proof suggests implicitly that every $X$ is the disjoint union of its connected components, i.e. that they are open - but why should this be true? For ordinary schemes this is true at least in the noetherian case.