A quasi-compact immersion with nonclosed image has a boundary specialization

Since $j(Z)$ is not closed, choose $t\in \overline{j(Z)}\setminus j(Z)$ and an affine open $V=\mathrm{Spec}R\subseteq T$ containing $t$; then $t$ lies in the closure in $V$ of $j(Z)\cap V$, because $V$ is an open neighbourhood of $t$, and $t\mathrm{\notin }j(Z)\cap V$.

By [F5] the morphism $j^{-1}(V)\to V$ is quasi-compact, and $V$ is affine, hence quasi-compact by [F4]; so $j^{-1}(V)$ is quasi-compact by [F2]. By [F3] it is covered by finitely many affine opens $Z_1,\dots ,Z_n$ with $Z_i=\mathrm{Spec}{A}_i$.

Now $j(Z)\cap V={\bigcup }_{i=1}^{n}j(Z_i)$, so the closure in $V$ of $j(Z)\cap V$ is the union of the finitely many closures of the $j(Z_i)$; as $t$ lies in that closure but in none of the $j(Z_i)$ (step 1.1), some index $i$ has $t\in \overline{j(Z_i)}\setminus j(Z_i)$. Fix such an $i$, write $\mathfrak{p}\subseteq R$ for the prime corresponding to $t$, and note $t\mathrm{\notin }j(Z_i)$.

For $f\in R$ with $f\mathrm{\notin }\mathfrak{p}$ one has $t\in D(f)$, so by [F7] the intersection $D(f)\cap j(Z_i)$ is nonempty; that set is the image of $\mathrm{Spec}((A_i{)}_f)$ under the composite $\mathrm{Spec}{A}_i\to V$, so $(A_i{)}_f\mathrm{\ne }0$, since a nonzero commutative ring has a prime ideal by [F9].

The rings $(A_i{)}_f$ for $f\mathrm{\notin }\mathfrak{p}$ form a filtered system with colimit $(A_i{)}_{\mathfrak{p}}$; since $1\mathrm{\ne }0$ in every $(A_i{)}_f$ by step 4.1, also $1\mathrm{\ne }0$ in the colimit, so $(A_i{)}_{\mathfrak{p}}\mathrm{\ne }0$. By [F9] the nonzero ring $(A_i{)}_{\mathfrak{p}}$ has a prime ideal ${\mathfrak{q}}^{\mathrm{\prime }}$, whose contraction $\mathfrak{q}\subseteq A_i$ satisfies $\mathfrak{q}\cap (R\setminus \mathfrak{p})=\mathrm{\varnothing }$, that is $\mathfrak{q}\cap R\subseteq \mathfrak{p}$.

Let $\eta$ be the point of $Z_i$ corresponding to $\mathfrak{q}$. By [F6] its image under $j$ is the prime $\mathfrak{q}\cap R\subseteq \mathfrak{p}$, and $\eta \in Z_i\subseteq Z$ so $j(\eta )\in j(Z_i)\subseteq j(Z)$. By [F8] the containment $\mathfrak{q}\cap R\subseteq \mathfrak{p}$ says exactly that $t\in \overline{\{\eta \}}$.

Steps 1.1, 3.1 and 6.1 produce $\eta \in j(Z)$ and $t\in \overline{j(Z)}\setminus j(Z)$ with $t\in \overline{\{\eta \}}$, which is the assertion. Only quasi-compactness of $j$, the affine basis of the topology and [F9] were used, the last being the exact use of the Axiom of Choice; the immersion hypothesis [F1] is not needed for this argument.