We apply Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated} to get open subspaces $U_p \subset X$, schemes $V_p$, and morphisms $f_p : V_p \to U_p$ with properties as stated. Note that $f_n : V_n \to U_n$ is an \'etale morphism of algebraic spaces whose restriction to the inverse image of $T_n = (V_n)_{red}$ is an isomorphism. Hence $f_n$ is an isomorphism, for example by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open}. In particular $U_n$ is a quasi-compact and separated scheme. Thus we can write $U_n = \lim U_{n, i}$ as a directed limit of schemes of finite type over $\mathbf{Z}$ with affine transition morphisms, see Limits, Proposition \ref{limits-proposition-approximate}. Thus, applying descending induction on $p$, we see that we have reduced to the problem posed in the following paragraph. \medskip\noindent Here we have $U \subset X$, $U = \lim U_i$, $Z \subset X$, and $f : V \to X$ with the following properties \begin{enumerate} \item $X$ is a quasi-compact and quasi-separated algebraic space, \item $V$ is a quasi-compact and separated scheme, \item $U \subset X$ is a quasi-compact open subspace, \item $(U_i, g_{ii'})$ is a directed inverse system of quasi-separated algebraic spaces of finite type over $\mathbf{Z}$ with affine transition morphisms whose limit is $U$, \item $Z \subset X$ is a closed subspace such that $|X| = |U| \amalg |Z|$, \item $f : V \to X$ is a surjective \'etale morphism such that $f^{-1}(Z) \to Z$ is an isomorphism. \end{enumerate} Problem: Show that the conclusion of the proposition holds for $X$. \medskip\noindent Note that $W = f^{-1}(U) \subset V$ is a quasi-compact open subscheme \'etale over $U$. Hence we may apply Lemmas \ref{lemma-descend-finite-presentation} and \ref{lemma-descend-etale} to find an index $0 \in I$ and an \'etale morphism $W_0 \to U_0$ of finite presentation whose base change to $U$ produces $W$. Setting $W_i = W_0 \times_{U_0} U_i$ we see that $W = \lim_{i \geq 0} W_i$. After increasing $0$ we may assume the $W_i$ are schemes, see Lemma \ref{lemma-limit-is-scheme}. Moreover, $W_i$ is of finite type over $\mathbf{Z}$. \medskip\noindent Apply Limits, Lemma \ref{limits-lemma-approximate} to $W = \lim_{i \geq 0} W_i$ and the inclusion $W \subset V$. Replace $I$ by the directed set $J$ found in that lemma. This allows us to write $V$ as a directed limit $V = \lim V_i$ of finite type schemes over $\mathbf{Z}$ with affine transition maps such that each $V_i$ contains $W_i$ as an open subscheme (compatible with transition morphisms). For each $i$ we can form the push out $$ \xymatrix{ W_i \ar[r] \ar[d]_\Delta & V_i \ar[d] \\ W_i \times_{U_i} W_i \ar[r] & R_i } $$ in the category of schemes. Namely, the left vertical and upper horizontal arrows are open immersions of schemes. In other words, we can construct $R_i$ as the glueing of $V_i$ and $W_i \times_{U_i} W_i$ along the common open $W_i$ (see Schemes, Section \ref{schemes-section-glueing-schemes}). Note that the \'etale projection maps $W_i \times_{U_i} W_i \to W_i$ extend to \'etale morphisms $s_i, t_i : R_i \to V_i$. It is clear that the morphism $j_i = (t_i, s_i) : R_i \to V_i \times V_i$ is an \'etale equivalence relation on $V_i$. Note that $W_i \times_{U_i} W_i$ is quasi-compact (as $U_i$ is quasi-separated and $W_i$ quasi-compact) and $V_i$ is quasi-compact, hence $R_i$ is quasi-compact. For $i \geq i'$ the diagram \begin{equation} \label{equation-cartesian} \vcenter{ \xymatrix{ R_i \ar[r] \ar[d]_{s_i} & R_{i'} \ar[d]^{s_{i'}} \\ V_i \ar[r] & V_{i'} } } \end{equation} is cartesian because $$ (W_{i'} \times_{U_{i'}} W_{i'}) \times_{U_{i'}} U_i = W_{i'} \times_{U_{i'}} U_i \times_{U_i} U_i \times_{U_{i'}} W_{i'} = W_i \times_{U_i} W_i. $$ Consider the algebraic space $X_i = V_i/R_i$ (see Spaces, Theorem \ref{spaces-theorem-presentation}). As $V_i$ is of finite type over $\mathbf{Z}$ and $R_i$ is quasi-compact we see that $X_i$ is quasi-separated and of finite type over $\mathbf{Z}$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated} and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-surjection-from-quasi-compact} and \ref{spaces-morphisms-lemma-finite-type-local}). As the construction of $R_i$ above is compatible with transition morphisms, we obtain morphisms of algebraic spaces $X_i \to X_{i'}$ for $i \geq i'$. The commutative diagrams $$ \xymatrix{ V_i \ar[r] \ar[d] & V_{i'} \ar[d] \\ X_i \ar[r] & X_{i'} } $$ are cartesian as (\ref{equation-cartesian}) is cartesian, see Groupoids, Lemma \ref{groupoids-lemma-criterion-fibre-product}. Since $V_i \to V_{i'}$ is affine, this implies that $X_i \to X_{i'}$ is affine, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-local}. Thus we can form the limit $X' = \lim X_i$ by Lemma \ref{lemma-directed-inverse-system-has-limit}. We claim that $X \cong X'$ which finishes the proof of the proposition. \medskip\noindent Proof of the claim. Set $R = \lim R_i$. By construction the algebraic space $X'$ comes equipped with a surjective \'etale morphism $V \to X'$ such that $$ V \times_{X'} V \cong R $$ (use Lemma \ref{lemma-directed-inverse-system-has-limit}). By construction $\lim W_i \times_{U_i} W_i = W \times_U W$ and $V = \lim V_i$ so that $R$ is the union of $W \times_U W$ and $V$ glued along $W$. Property (6) implies the projections $V \times_X V \to V$ are isomorphisms over $f^{-1}(Z) \subset V$. Hence the scheme $V \times_X V$ is the union of the opens $\Delta_{V/X}(V)$ and $W \times_U W$ which intersect along $\Delta_{W/X}(W)$. We conclude that there exists a unique isomorphism $R \cong V \times_X V$ compatible with the projections to $V$. Since $V \to X$ and $V \to X'$ are surjective \'etale we see that $$ X = V/ V \times_X V = V/R = V/V \times_{X'} V = X' $$ by Spaces, Lemma \ref{spaces-lemma-space-presentation} and we win.