= Solution
{c}
\b[The immersions and gluing.] For $p>0$, the <prime ideals> of the <localization> $\mathbb Z_{(p)}$ are $(0)$ and $(p)\mathbb Z_{(p)}$. Inverting $p$ gives $\mathbb Z_{(p)}[1/p]=\mathbb Q$, so
$$
\boxed{\psi_p:\operatorname{Spec}\mathbb Q\xrightarrow{\ \sim\ }D(p)\hookrightarrow U_p}
$$
is an <open immersion> sending the <generic point> to the <generic point>. For $p=0$, $\mathbb Z_{(0)}=\mathbb Q$, and take $\psi_0$ to be the identity. Identify each pair of these <open subschemes> using the identity of $\operatorname{Spec}\mathbb Q$. The identities compose to the identity on every triple overlap, so <gluing of schemes along open subschemes> applies. No finiteness of the family is required.
The resulting <scheme> $Y$ has one common <generic point> $\eta$ and a distinct <closed point> $y_p$ for every positive <prime number> $p$. Its <affine open subschemes> $U_p=\{\eta,y_p\}$, for $p>0$, intersect pairwise in $U_0=\{\eta\}$. A subset $W\subseteq Y$ is open precisely when its intersection with every $U_p$ is open. In the two-point <topological space> $U_p$, the open subsets are $\varnothing$, $\{\eta\}$ and $U_p$. Consequently
$$
\boxed{\text{The nonempty opens of }Y\text{ are exactly }W_S=\{\eta\}\cup\{y_p:p\in S\},}
$$
where $S$ is any subset of the positive <prime numbers>, including the empty subset.
\b[The canonical morphism.] The <ring homomorphisms> $\mathbb Z\to\mathbb Z_{(p)}$ give <morphisms of schemes> $U_p\to\operatorname{Spec}\mathbb Z$ agreeing on the overlaps, hence a <morphism of schemes> $f:Y\to\operatorname{Spec}\mathbb Z$. By uniqueness of the <structure morphism>, this is the required map. Contraction of <prime ideals> gives
$$
\boxed{f(\eta)=(0),\qquad f(y_p)=(p).}
$$
These are all points of $\operatorname{Spec}\mathbb Z$, each exactly once, so $f$ is a <bijection>. It is continuous because its restrictions to the <open cover> $(U_p)$ are continuous; explicitly, every nonempty <open subset> of $\operatorname{Spec}\mathbb Z$ is the complement of a finite collection of <closed points>, whose inverse image is one of the $W_S$.
\b[Sections on every open.] For any $S$, the <sheaf gluing axiom> on the cover of $W_S$ by $U_0$ and the $U_p$ with $p\in S$ identifies
$$
\boxed{\Gamma(W_S,\mathcal O_Y)=\bigcap_{p\in S}\mathbb Z_{(p)}\ \subseteq\mathbb Q,}
$$
where the intersection over the empty set is $\mathbb Q$. Indeed, a section on $U_p$ is an element of $\mathbb Z_{(p)}$; restrictions to every overlap are its inclusion into $\mathbb Q$. Compatibility means that all these elements are the same rational number. In lowest terms $a/b$, this rational number lies in the intersection exactly when no <prime number> in $S$ divides $b$. Equivalently, this is the <integer localization at a set of primes> obtained by inverting all positive <prime numbers> outside $S$.
Now let $V\subseteq\operatorname{Spec}\mathbb Z$ be nonempty and open. Its finite complement consists of $(p_1),\ldots,(p_r)$; put $N=p_1\cdots p_r$, with $N=1$ for an empty complement. Then $V=D(N)$ and
$$
\Gamma(V,\mathcal O_{\operatorname{Spec}\mathbb Z})=\mathbb Z[1/N]
=\bigcap_{p\nmid N}\mathbb Z_{(p)}
=\Gamma(f^{-1}V,\mathcal O_Y).
$$
The equality follows by the denominator criterion above. The map $f^\#$ is this same equality of subrings of $\mathbb Q$, so it is a <ring isomorphism>. Over the empty <open subset> both rings of sections are the <zero ring>. These identifications respect every restriction map. Thus the precise comparison of <structure sheaves> is
$$
\boxed{\mathcal O_{\operatorname{Spec}\mathbb Z}\xrightarrow{\ \sim\ }f_*\mathcal O_Y.}
$$
Here “over every open set” refers to an <open subset> $V$ of the target and its inverse image. The preceding intersection formula additionally describes the sections on every <open subset> of $Y$, including ones whose images are not open in the target.
\b[Topology and finiteness.] The singleton $\{\eta\}=U_0$ is open in $Y$, but $\{(0)\}$ is not open in $\operatorname{Spec}\mathbb Z$: a nonempty <open subset> there contains all but finitely many of the infinitely many <closed points>. Therefore $f$ is not a <homeomorphism>, despite being a continuous <bijection> and giving the indicated <isomorphism> of <structure sheaves> after <direct image>.
The <open cover> $(U_p)_{p>0}$ of $Y$ has no finite subcover, because $y_q$ belongs only to $U_q$. Thus $Y$ is not a <quasi-compact scheme>, and part (a) rules out an <affine scheme>. Each $\mathbb Z_{(p)}$ is a <Noetherian ring>, being a <localization> of the <Noetherian ring> $\mathbb Z$. Hence the $U_p$ form an <affine open cover> by spectra of <Noetherian rings>, proving that $Y$ is a <locally Noetherian scheme>. A <Noetherian scheme> is a <quasi-compact scheme>, so
$$
\boxed{Y\text{ is locally Noetherian, not quasi-compact, not affine, and not Noetherian}.}
$$
\Image[/past-exam-of-the-introduction-to-schemes-course-of-the-university-of-oxford/2021-glued-topology.png]
{title=Gluing the local spectra retains every closed point but permits more open subsets}
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