Affine neighbourhood containing component generic points

Since $X=Z_1\cup \cdots \cup Z_r$, the point $x$ lies in at least one component. Reindex the components so that $x\in Z_1,\dots ,Z_{r^{\mathrm{\prime }}}$ and $x\mathrm{\notin }{Z}_{r^{\mathrm{\prime }}+1},\dots ,Z_r$ for some $0\le r^{\mathrm{\prime }}\le r$. If $r^{\mathrm{\prime }}=r$ there are no indices in the second range and the set $X$ below is all of $X$.

Every $Z_i$ equals $\overline{\{{\eta }_i\}}$ and is closed in $X$, by [F3]. Hence $V:=X\setminus (Z_{r^{\mathrm{\prime }}+1}\cup \cdots \cup Z_r)$ is an open subset of $X$ containing $x$, and the union displayed is a finite union of closed subsets.

Choose an affine open $W_0\subseteq X$ with $x\in W_0$, possible by [F1]. Then $W_0\subseteq \mathrm{Spec}(A_0)$ for $A_0=\mathrm{\Gamma }(W_0,{\mathcal{O}}_X)$ under the affine structure, $W_0\cap V$ is an open subset of the affine scheme $W_0$ containing $x$, and by [F2] there is $f\in A_0$ with $x\in W:=D(f)\subseteq W_0\cap V$. Thus $W$ is an affine open subscheme of $X$ with $x\in W$ and $W\cap Z_i=\mathrm{\varnothing }$ for $i>r^{\mathrm{\prime }}$.

Let $i\le r^{\mathrm{\prime }}$. Then $W\cap Z_i$ is a nonempty open subset of the irreducible space $Z_i$, since $x\in W\cap Z_i$. If the generic point ${\eta }_i$ did not lie in $W\cap Z_i$, then $Z_i\setminus W$ would be a closed subset of $Z_i$ containing ${\eta }_i$; as $Z_i=\overline{\{{\eta }_i\}}$, this forces $Z_i\setminus W\supseteq \overline{\{{\eta }_i\}}=Z_i$, contradicting $W\cap Z_i\mathrm{\ne }\mathrm{\varnothing }$. Hence ${\eta }_i\in W$ for every $i\le r^{\mathrm{\prime }}$.

For $i>r^{\mathrm{\prime }}$ the generic point ${\eta }_i$ does not lie in any $Z_k$ with $k\mathrm{\ne }i$: otherwise $Z_i=\overline{\{{\eta }_i\}}\subseteq Z_k$, and since $Z_i$ is an irreducible component contained in the irreducible subset $Z_k$, maximality [F4] forces $Z_i=Z_k$, contrary to the components being indexed distinctly. Hence $X\setminus {\bigcup }_{k\mathrm{\ne }i}{Z}_k$ is an open neighbourhood of ${\eta }_i$; choosing an affine open of $X$ inside it and then a distinguished open inside the resulting affine scheme as in step 3.1, we obtain an affine open subscheme $V_i$ with ${\eta }_i\in V_i$ and $V_i\cap Z_k=\mathrm{\varnothing }$ for all $k\mathrm{\ne }i$.

The opens $V_i$ from step 4.2 are already pairwise disjoint. Indeed, $V_i\subseteq X\setminus {\bigcup }_{l\mathrm{\ne }i}{Z}_l$, and every point of $X$ belongs to one of the components $Z_l$, so $V_i\subseteq Z_i$. For $k\mathrm{\ne }i$, step 4.2 gives $V_k\cap Z_i=\mathrm{\varnothing }$; hence $V_i\cap V_k=\mathrm{\varnothing }$. Also, for $i>r^{\mathrm{\prime }}$ step 3.1 gives $W\cap Z_i=\mathrm{\varnothing }$, so $W\cap V_i=\mathrm{\varnothing }$.

For each $i>r^{\mathrm{\prime }}$ put $U_i^{\mathrm{\prime }}:=V_i$. This is an affine open containing ${\eta }_i$ by step 4.2. By step 5.1 these opens are pairwise disjoint and each is disjoint from $W$; no further shrinking or openness of a set difference is needed.

Put $U:=W\cup {\bigcup }_{i>r^{\mathrm{\prime }}}{U}_i^{\mathrm{\prime }}$. This is an open subscheme of $X$. It contains $x$ and all ${\eta }_1,\dots ,{\eta }_r$: the points ${\eta }_i$ with $i\le r^{\mathrm{\prime }}$ lie in $W$ by step 4.1, and each ${\eta }_i$ with $i>r^{\mathrm{\prime }}$ lies in $U_i^{\mathrm{\prime }}$ by step 6.1. The pieces are pairwise disjoint open subschemes: $W\cap U_i^{\mathrm{\prime }}=\mathrm{\varnothing }$ for every $i>r^{\mathrm{\prime }}$ by step 6.1 and $U_i^{\mathrm{\prime }}\cap U_k^{\mathrm{\prime }}=\mathrm{\varnothing }$ for $i\mathrm{\ne }k$ by step 6.1, and each piece is affine.

The open subscheme $U$ is affine and $\mathrm{\Gamma }(U,{\mathcal{O}}_X)\cong \mathrm{\Gamma }(W,{\mathcal{O}}_X)\times {\prod }_{i>r^{\mathrm{\prime }}}\mathrm{\Gamma }(U_i^{\mathrm{\prime }},{\mathcal{O}}_X)$. Indeed, $U$ is the disjoint union of the affine open subschemes $W$ and $U_i^{\mathrm{\prime }}$ ($i>r^{\mathrm{\prime }}$), so by [F5] the restriction maps identify $\mathrm{\Gamma }(U,{\mathcal{O}}_X)$ with the product $A:=\mathrm{\Gamma }(W,{\mathcal{O}}_X)\times {\prod }_{i>r^{\mathrm{\prime }}}\mathrm{\Gamma }(U_i^{\mathrm{\prime }},{\mathcal{O}}_X)$, the product being taken over the empty set when $r^{\mathrm{\prime }}=r$. Let $e_0,e_i$ ($i>r^{\mathrm{\prime }}$) be the idempotents of $A$; by [F7] the spectrum $\mathrm{Spec}A$ is the disjoint union of the clopen pieces $D(e_0)=\mathrm{Spec}\mathrm{\Gamma }(W,{\mathcal{O}}_X)$ and $D(e_i)=\mathrm{Spec}\mathrm{\Gamma }(U_i^{\mathrm{\prime }},{\mathcal{O}}_X)$, and the morphisms induced by the projections are isomorphisms onto these pieces. The inverse ring isomorphism $A\to \mathrm{\Gamma }(U,{\mathcal{O}}_X)$ of [F5] corresponds by [F6] to a morphism $h:U\to \mathrm{Spec}A$; for each piece, restriction to that piece corresponds by the compatibility in [F6] to the composite of the inverse isomorphism with the projection, so $h$ restricts to an isomorphism $W\to D(e_0)$ on $W$ and to an isomorphism $U_i^{\mathrm{\prime }}\to D(e_i)$ on $U_i^{\mathrm{\prime }}$. Since the sources of these restrictions cover $U$, the targets cover $\mathrm{Spec}A$, and on each piece the structure-sheaf map is an isomorphism, $h$ is a homeomorphism and induces an isomorphism of structure sheaves; hence $h$ is an isomorphism of schemes and $U\cong \mathrm{Spec}A$ is affine.

By steps 7.1 and 8.1 the subscheme $U$ is an affine open subscheme of $X$ containing $x$ and all generic points ${\eta }_1,\dots ,{\eta }_r$ of the components of $X$. If $X=\mathrm{\varnothing }$ then $r=0$ and there is no point $x$, so the assertion is vacuous. The quasi-separatedness hypothesis is retained though the argument above only used that affine opens form a basis of the topology, that finitely many pairwise disjoint affine opens may be adjoined, and that the union is affine. No choice principle is used: all selections are made inside the single affine charts $W_0$ and $V_i$ by [F2], and the family of components is finite and given.