Finite-type schemes have finitely many clopen connected components

By [F2], take a finite affine cover $X={\bigcup }_{i=1}^m\mathrm{Spec}{R}_i$ with each $R_i$ a finitely generated $k$-algebra. A field has only the zero and unit ideals: an ideal containing $a\mathrm{\ne }0$ contains $a^{-1}a=1$ by [F1]. The zero ideal is generated by $0$, and the unit ideal by $1$, by [F10]. Thus both are finitely generated, so [F3] makes $k$ and then each $R_i$ Noetherian. By [F4], $X$ is a Noetherian scheme. Its underlying space is Noetherian too: a descending chain of closed subsets restricts to a stationary chain on each affine by [F3], and the maximum of the finitely many stationarity indices works on the whole cover. The empty scheme uses the empty cover and is immediate.

By [F4] and step 1.1, write $\mathrm{\mid }X\mathrm{\mid }={\bigcup }_{i=1}^n{C}_i$ with irreducible closed $C_i$. Each $C_i$ is connected, since a separation would express it as two proper relatively closed subsets. Declare indices equivalent when joined by a finite sequence of indices with successive $C_i$ intersecting. Each equivalence class gives a connected union: in any separation, every connected $C_i$ lies on one side and intersections force all members of the sequence onto the same side. These class unions are disjoint closed subsets covering $\mathrm{\mid }X\mathrm{\mid }$, finite in number; each is open because its complement is a finite union of the other closed unions. A connected subset meeting one cannot meet its complement, since their intersections would separate it. Thus these unions are exactly the connected components, which are finite and clopen. For empty $X$ there are no components.

Both function sets in the statement carry abelian group structures under pointwise addition: the zero function is locally constant, negation preserves constancy on each open neighbourhood, and the sum of two locally constant functions is constant on the intersection of open neighbourhoods on which each is constant, by [F9] and [F13]. On all functions on the component set these operations are unrestricted. The group equations hold pointwise by [F11]. Every value fiber of a locally constant function is open, since [F13] gives an open neighbourhood contained in that fiber at each of its points, and [F9] permits their union. Such a function is therefore constant on each connected component: if it took different values on a component, one nonempty value fiber and the union of its other value fibers would separate the component. Conversely, any function on the components defines a locally constant function on $\mathrm{\mid }X\mathrm{\mid }$, since each point lies in exactly one component and each component is an open neighbourhood of each of its points by step 2.1. Taking the unique value on each component and assigning component values back to points are mutually inverse constructions; they preserve pointwise addition, hence give a group isomorphism by [F12]. By [F5], a scheme morphism has a continuous underlying map, and a connected subset has connected image: a separation of the image would pull back to a separation of the subset, using the relative topologies in [F9] and the open-neighbourhood criterion in [F5]. Indeed the preimage of an open set is the union of the open neighbourhoods supplied at its preimage points. Thus precomposition preserves local constancy, and the morphism sends each component into a unique component. Precomposition on the two function groups commutes with the displayed identification: both composites assign to a source component the value on the component containing its image. For empty $X$ both function sets consist only of the empty function, which is their zero element. This proves the entire statement.