hcju/stacksqa huggingface dataset:
If (1) holds then the morphisms $U_i \to X_i \to X$ are \'etale (combine Morphisms, Lemma \ref{morphisms-lemma-composition-etale} and Spaces, Lemmas \ref{spaces-lemma-composition-representable-transformations-property} and \ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property} ). Moreover, as $U_i \times_X U_i = U_i \times_{X_i} U_i$, both projections $U_i \times_X U_i \to U_i$ are quasi-compact. \medskip\noindent If (2) holds then let $X_i \subset X$ be the open subspace corresponding to the image of the open map $|U_i| \to |X|$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-image-open}. The morphisms $U_i \to X_i$ are surjective. Hence $U_i \to X_i$ is surjective \'etale, and the projections $U_i \times_{X_i} U_i \to U_i$ are quasi-compact, because $U_i \times_{X_i} U_i = U_i \times_X U_i$. Thus by Spaces, Lemma \ref{spaces-lemma-representable-morphisms-spaces-property} the morphisms $U_i \to X_i$ are quasi-compact.
The implication $(\gamma) \Leftrightarrow (\alpha) + (\beta)$ is immediate. The implications in the diamond on the left are clear from the definitions. \medskip\noindent Assume $(\zeta)$, i.e., that $X$ is Zariski locally quasi-separated. Then $(\epsilon)$ holds by Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated-quasi-compact-pieces}. \medskip\noindent Assume $(\epsilon)$. By Lemma \ref{lemma-characterize-very-reasonable} there exists a Zariski open covering $X = \bigcup X_i$ such that for each $i$ there exists a scheme $U_i$ and a quasi-compact surjective \'etale morphism $U_i \to X_i$. Choose an $i$ and an affine open subscheme $W \subset U_i$. It suffices to show that $W \to X$ has universally bounded fibres, since then the family of all these morphisms $W \to X$ covers $X$. To do this we consider the diagram $$ \xymatrix{ W \times_X U_i \ar[r]_-p \ar[d]_q & U_i \ar[d] \\ W \ar[r] & X } $$ Since $W \to X$ factors through $X_i$ we see that $W \times_X U_i = W \times_{X_i} U_i$, and hence $q$ is quasi-compact. Since $W$ is affine this implies that the scheme $W \times_X U_i$ is quasi-compact. Thus we may apply Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded} and we conclude that $p$ has universally bounded fibres. From Lemma \ref{lemma-descent-universally-bounded} we conclude that $W \to X$ has universally bounded fibres as well. \medskip\noindent Assume $(\delta)$. Let $U$ be an affine scheme, and let $U \to X$ be an \'etale morphism. By assumption the fibres of the morphism $U \to X$ are universally bounded. Thus also the fibres of both projections $R = U \times_X U \to U$ are universally bounded, see Lemma \ref{lemma-base-change-universally-bounded}. And by Lemma \ref{lemma-composition-universally-bounded} also the fibres of $R \to X$ are universally bounded. Hence for any $x \in X$ the fibres of $|U| \to |X|$ and $|R| \to |X|$ over $x$ are finite, see Lemma \ref{lemma-universally-bounded-finite-fibres}. In other words, the equivalent conditions of Lemma \ref{lemma-UR-finite-above-x} hold. This proves that $(\delta) \Rightarrow (\gamma)$.
Let $X$ be an algebraic space over $S$, and let $X = \bigcup X_i$ is a Zariski open covering such that each $X_i$ has $\mathcal{P}$. \medskip\noindent The case $\mathcal{P} = (\alpha)$. The condition $(\alpha)$ for $X_i$ means that for every $x \in |X_i|$ and every affine scheme $U$, and \'etale morphism $\varphi : U \to X_i$ the fibre of $\varphi : |U| \to |X_i|$ over $x$ is finite. Consider $x \in X$, an affine scheme $U$ and an \'etale morphism $U \to X$. Since $X = \bigcup X_i$ is a Zariski open covering there exits a finite affine open covering $U = U_1 \cup \ldots \cup U_n$ such that each $U_j \to X$ factors through some $X_{i_j}$. By assumption the fibres of $|U_j | \to |X_{i_j}|$ over $x$ are finite for $j = 1, \ldots, n$. Clearly this means that the fibre of $|U| \to |X|$ over $x$ is finite. This proves the result for $(\alpha)$. \medskip\noindent The case $\mathcal{P} = (\beta)$. The condition $(\beta)$ for $X_i$ means that every $x \in |X_i|$ is represented by a monomorphism from the spectrum of a field towards $X_i$. Hence the same follows for $X$ as $X_i \to X$ is a monomorphism and $X = \bigcup X_i$. \medskip\noindent The case $\mathcal{P} = (\gamma)$. Note that $(\gamma) = (\alpha) + (\beta)$ by Lemma \ref{lemma-bounded-fibres} hence the lemma for $(\gamma)$ follows from the cases treated above. \medskip\noindent The case $\mathcal{P} = (\delta)$. The condition $(\delta)$ for $X_i$ means there exist schemes $U_{ij}$ and \'etale morphisms $U_{ij} \to X_i$ with universally bounded fibres which cover $X_i$. These schemes also give an \'etale surjective morphism $\coprod U_{ij} \to X$ and $U_{ij} \to X$ still has universally bounded fibres. \medskip\noindent The case $\mathcal{P} = (\epsilon)$. The condition $(\epsilon)$ for $X_i$ means we can find a set $J_i$ and morphisms $\varphi_{ij} : U_{ij} \to X_i$ such that each $\varphi_{ij}$ is \'etale, both projections $U_{ij} \times_{X_i} U_{ij} \to U_{ij}$ are quasi-compact, and $\coprod_{j \in J_i} U_{ij} \to X_i$ is surjective. In this case the compositions $U_{ij} \to X_i \to X$ are \'etale (combine Morphisms, Lemmas \ref{morphisms-lemma-composition-etale} and \ref{morphisms-lemma-open-immersion-etale} and Spaces, Lemmas \ref{spaces-lemma-composition-representable-transformations-property} and \ref{spaces-lemma-morphism-schemes-gives-representable-transformation-property} ). Since $X_i \subset X$ is a subspace we see that $U_{ij} \times_{X_i} U_{ij} = U_{ij} \times_X U_{ij}$, and hence the condition on fibre products is preserved. And clearly $\coprod_{i, j} U_{ij} \to X$ is surjective. Hence $X$ satisfies $(\epsilon)$. \medskip\noindent The case $\mathcal{P} = (\zeta)$. The condition $(\zeta)$ for $X_i$ means that $X_i$ is Zariski locally quasi-separated. It is immediately clear that this means $X$ is Zariski locally quasi-separated. \medskip\noindent For $(\theta)$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme}.
Assume $f : X \to Y$ is a representable morphism of algebraic spaces, and assume that $Y$ has $\mathcal{P}$. Let $x \in |X|$, and set $y = f(x) \in |Y|$. \medskip\noindent The case $\mathcal{P} = (\beta)$. Condition $(\beta)$ for $Y$ means there exists a monomorphism $\Spec(k) \to Y$ representing $y$. The fibre product $X_y = \Spec(k) \times_Y X$ is a scheme, and $x$ corresponds to a point of $X_y$, i.e., to a monomorphism $\Spec(k') \to X_y$. As $X_y \to X$ is a monomorphism also we see that $x$ is represented by the monomorphism $\Spec(k') \to X_y \to X$. In other words $(\beta)$ holds for $X$. \medskip\noindent The case $\mathcal{P} = (\gamma)$. Since $(\gamma) \Rightarrow (\beta)$ we have seen in the preceding paragraph that $y$ and $x$ can be represented by monomorphisms as in the following diagram $$ \xymatrix{ \Spec(k') \ar[r]_-x \ar[d] & X \ar[d] \\ \Spec(k) \ar[r]^-y & Y } $$ Also, by definition of property $(\gamma)$ via Lemma \ref{lemma-UR-finite-above-x} (2) there exist schemes $V_i$ and \'etale morphisms $V_i \to Y$ such that $\coprod V_i \to Y$ is surjective and for each $i$, setting $R_i = V_i \times_Y V_i$ the fibres of both $$ |V_i| \longrightarrow |Y| \quad\text{and}\quad |R_i| \longrightarrow |Y| $$ over $y$ are finite. This means that the schemes $(V_i)_y$ and $(R_i)_y$ are finite schemes over $y = \Spec(k)$. As $X \to Y$ is representable, the fibre products $U_i = V_i \times_Y X$ are schemes. The morphisms $U_i \to X$ are \'etale, and $\coprod U_i \to X$ is surjective. Finally, for each $i$ we have $$ (U_i)_x = (V_i \times_Y X)_x = (V_i)_y \times_{\Spec(k)} \Spec(k') $$ and $$ (U_i \times_X U_i)_x = \left((V_i \times_Y X) \times_X (V_i \times_Y X)\right)_x = (R_i)_y \times_{\Spec(k)} \Spec(k') $$ hence these are finite over $k'$ as base changes of the finite schemes $(V_i)_y$ and $(R_i)_y$. This implies that $(\gamma)$ holds for $X$, again via the second condition of Lemma \ref{lemma-UR-finite-above-x}. \medskip\noindent The case $\mathcal{P} = (\delta)$. Let $V \to Y$ be an \'etale morphism with $V$ an affine scheme. Since $Y$ has property $(\delta)$ this morphism has universally bounded fibres. By Lemma \ref{lemma-base-change-universally-bounded} the base change $V \times_Y X \to X$ also has universally bounded fibres. Hence the first part of Lemma \ref{lemma-U-universally-bounded} applies and we see that $Y$ also has property $(\delta)$. \medskip\noindent The case $\mathcal{P} = (\epsilon)$. We will repeatedly use Spaces, Lemma \ref{spaces-lemma-base-change-representable-transformations-property}. Let $V_i \to Y$ be as in Lemma \ref{lemma-characterize-very-reasonable} (2). Set $U_i = X \times_Y V_i$. The morphisms $U_i \to X$ are \'etale, and $\coprod U_i \to X$ is surjective. Because $U_i \times_X U_i = X \times_Y (V_i \times_Y V_i)$ we see that the projections $U_i \times_Y U_i \to U_i$ are base changes of the projections $V_i \times_Y V_i \to V_i$, and so quasi-compact as well. Hence $X$ satisfies Lemma \ref{lemma-characterize-very-reasonable} (2). \medskip\noindent The case $\mathcal{P} = (\theta)$. In this case the result is Categories, Lemma \ref{categories-lemma-representable-over-representable}.
We sketch the proof. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover} we have $X = U/R$ with $U$ affine. In this case, reasonable means $U \to X$ is universally bounded. Hence there exists an integer $N$ such that the ``fibres'' of $U \to X$ have degree at most $N$, see Definition \ref{definition-universally-bounded}. Denote $s, t : R \to U$ and $c : R \times_{s, U, t} R \to R$ the groupoid structural maps. \medskip\noindent Claim: for every quasi-compact open $A \subset R$ there exists an open $R' \subset R$ such that \begin{enumerate} \item $A \subset R'$, \item $R'$ is quasi-compact, and \item $(U, R', s|_{R'}, t|_{R'}, c|_{R' \times_{s, U, t} R'})$ is a groupoid scheme. \end{enumerate} Note that $e : U \to R$ is open as it is a section of the \'etale morphism $s : R \to U$, see \'Etale Morphisms, Proposition \ref{etale-proposition-properties-sections}. Moreover $U$ is affine hence quasi-compact. Hence we may replace $A$ by $A \cup e(U) \subset R$, and assume that $A$ contains $e(U)$. Next, we define inductively $A^1 = A$, and $$ A^n = c(A^{n - 1} \times_{s, U, t} A) \subset R $$ for $n \geq 2$. Arguing inductively, we see that $A^n$ is quasi-compact for all $n \geq 2$, as the image of the quasi-compact fibre product $A^{n - 1} \times_{s, U, t} A$. If $k$ is an algebraically closed field over $S$, and we consider $k$-points then $$ A^n(k) = \left\{(u, u') \in U(k) : \begin{matrix} \text{there exist } u = u_1, u_2, \ldots, u_n \in U(k)\text{ with} \\ (u_i , u_{i + 1}) \in A \text{ for all }i = 1, \ldots, n - 1. \end{matrix} \right\} $$ But as the fibres of $U(k) \to X(k)$ have size at most $N$ we see that if $n > N$ then we get a repeat in the sequence above, and we can shorten it proving $A^N = A^n$ for all $n \geq N$. This implies that $R' = A^N$ gives a groupoid scheme $(U, R', s|_{R'}, t|_{R'}, c|_{R' \times_{s, U, t} R'})$, proving the claim above. \medskip\noindent Consider the map of sheaves on $(\Sch/S)_{fppf}$ $$ \colim_{R' \subset R} U/R' \longrightarrow U/R $$ where $R' \subset R$ runs over the quasi-compact open subschemes of $R$ which give \'etale equivalence relations as above. Each of the quotients $U/R'$ is an algebraic space (see Spaces, Theorem \ref{spaces-theorem-presentation}). Since $R'$ is quasi-compact, and $U$ affine the morphism $R' \to U \times_{\Spec(\mathbf{Z})} U$ is quasi-compact, and hence $U/R'$ is quasi-separated. Finally, if $T$ is a quasi-compact scheme, then $$ \colim_{R' \subset R} U(T)/R'(T) \longrightarrow U(T)/R(T) $$ is a bijection, since every morphism from $T$ into $R$ ends up in one of the open subrelations $R'$ by the claim above. This clearly implies that the colimit of the sheaves $U/R'$ is $U/R$. In other words the algebraic space $X = U/R$ is the colimit of the quasi-separated algebraic spaces $U/R'$.
Assume the pair $(X, x)$ satisfies the equivalent conditions for Lemma \ref{lemma-U-finite-above-x}. Let $U$ be a scheme, $U \to X$ \'etale, and let $u, u' \in |U|$ map to $x$ of $|X|$, and $u' \leadsto u$. We may and do replace $U$ by an affine neighbourhood of $u$. Let $t, s : R = U \times_X U \to U$ be the \'etale projection maps. \medskip\noindent Pick a point $r \in R$ with $t(r) = u$ and $s(r) = u'$. This is possible by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-presentation}. Because generalizations lift along the \'etale morphism $t$ (Remark \ref{remark-recall}) we can find a specialization $r' \leadsto r$ with $t(r') = u'$. Set $u'' = s(r')$. Then $u'' \leadsto u'$. Thus we may repeat and find $r'' \leadsto r'$ with $t(r'') = u''$. Set $u''' = s(r'')$, and so on. Here is a picture: $$ \xymatrix{ & r'' \ar[rd]^s \ar[ld]_t \ar@{~>}[d] & \\ u'' \ar@{~>}[d] & r' \ar[rd]^s \ar[ld]_t \ar@{~>}[d] & u''' \ar@{~>}[d] \\ u' \ar@{~>}[d] & r \ar[rd]^s \ar[ld]_t & u'' \ar@{~>}[d] \\ u & & u' } $$ In Remark \ref{remark-recall} we have seen that there are no specializations among points in the fibres of the \'etale morphism $s$. Hence if $u^{(n + 1)} = u^{(n)}$ for some $n$, then also $r^{(n)} = r^{(n - 1)}$ and hence also (by taking $t$) $u^{(n)} = u^{(n - 1)}$. This then forces the whole tower to collapse, in particular $u = u'$. Thus we see that if $u \not = u'$, then all the specializations are strict and $\{u, u', u'', \ldots\}$ is an infinite set of points in $U$ which map to the point $x$ in $|X|$. As we chose $U$ affine this contradicts the second part of Lemma \ref{lemma-U-finite-above-x}, as desired.
We may replace $U$ by an affine open neighbourhood of $u$. Hence we may assume that $U$ is affine. As $x$ is in the image of the open map $|U| \to |X|$, so is $x'$. Thus we may replace $X$ by the Zariski open subspace corresponding to the image of $|U| \to |X|$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-image-open}. In other words we may assume that $U \to X$ is surjective and \'etale. Let $s, t : R = U \times_X U \to U$ be the projections. By our assumption that $(X, x')$ satisfies the equivalent conditions of Lemma \ref{lemma-UR-finite-above-x} we see that the fibres of $|U| \to |X|$ and $|R| \to |X|$ over $x'$ are finite. Say $\{u'_1, \ldots, u'_n\} \subset U$ and $\{r'_1, \ldots, r'_m\} \subset R$ form the complete inverse image of $\{x'\}$. Consider the closed sets $$ T = \overline{\{u'_1\}} \cup \ldots \cup \overline{\{u'_n\}} \subset |U|, \quad T' = \overline{\{r'_1\}} \cup \ldots \cup \overline{\{r'_m\}} \subset |R|. $$ Trivially we have $s(T') \subset T$. Because $R$ is an equivalence relation we also have $t(T') = s(T')$ as the set $\{r_j'\}$ is invariant under the inverse of $R$ by construction. Let $w \in T$ be any point. Then $u'_i \leadsto w$ for some $i$. Choose $r \in R$ with $s(r) = w$. Since generalizations lift along $s : R \to U$, see Remark \ref{remark-recall}, we can find $r' \leadsto r$ with $s(r') = u_i'$. Then $r' = r'_j$ for some $j$ and we conclude that $w \in s(T')$. Hence $T = s(T') = t(T')$ is an $|R|$-invariant closed set in $|U|$. This means $T$ is the inverse image of a closed (!) subset $T'' = \varphi(T)$ of $|X|$, see Properties of Spaces, Lemmas \ref{spaces-properties-lemma-points-presentation} and \ref{spaces-properties-lemma-topology-points}. Hence $T'' = \overline{\{x'\}}$. Thus $T$ contains some point $u_1$ mapping to $x$ as $x \in T''$. I.e., we see that for some $i$ there exists a specialization $u'_i \leadsto u_1$ which maps to the given specialization $x' \leadsto x$. \medskip\noindent To finish the proof, choose a point $r \in R$ such that $s(r) = u$ and $t(r) = u_1$ (using Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}). As generalizations lift along $t$, and $u'_i \leadsto u_1$ we can find a specialization $r' \leadsto r$ such that $t(r') = u'_i$. Set $u' = s(r')$. Then $u' \leadsto u$ and $\varphi(u') = x'$ as desired.
(The parenthetical statement holds by the definition of decent spaces and the implications between the different separation conditions mentioned in Section \ref{section-reasonable-decent}.) Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose $v \in V$ mapping to $y$. Then we see that it suffices to prove the lemma for $V \to X$. Thus we may assume $Y$ is a scheme. Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. Choose $u \in U$ mapping to $x$. By Lemma \ref{lemma-specialization} we may choose $u' \leadsto u$ mapping to $x'$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} we may choose $z \in U \times_X Y$ mapping to $y$ and $u$. Thus we reduce to the case of the flat morphism of schemes $U \times_X Y \to U$ which is Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}.
Let $n$ be an integer bounding the degrees of the fibres of $W \to X$. Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. Apply More on Morphisms, Lemma \ref{more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded} to $W \times_X U \to U$. We obtain closed subsets $$ \emptyset = Y_{-1} \subset Y_0 \subset Y_1 \subset Y_2 \subset \ldots \subset Y_n = U $$ characterized by the property stated in the lemma for the morphism $W \times_X U \to U$. Clearly, the formation of these closed subsets commutes with base change. Setting $R = U \times_X U$ with projection maps $s, t : R \to U$ we conclude that $$ s^{-1}(Y_r) = t^{-1}(Y_r) $$ as closed subsets of $R$. In other words the closed subsets $Y_r \subset U$ are $R$-invariant. This means that $|Y_r|$ is the inverse image of a closed subset $Z_r \subset |X|$. Denote $Z_r \subset X$ also the reduced induced algebraic space structure, see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. \medskip\noindent Let $g : T \to X$ be a morphism of algebraic spaces. Choose a scheme $V$ and a surjective \'etale morphism $V \to T$. To prove the final assertion of the lemma it suffices to prove the assertion for the composition $V \to X$ (by our definition of finite locally free morphisms, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-finite-locally-free}). Similarly, the morphism of schemes $W \times_X V \to V$ is finite locally free of degree $r$ if and only if the morphism of schemes $$ W \times_X (U \times_X V) \longrightarrow U \times_X V $$ is finite locally free of degree $r$ (see Descent, Lemma \ref{descent-lemma-descending-property-finite-locally-free}). By construction this happens if and only if $|U \times_X V| \to |U|$ maps into $|Y_r|$, which is true if and only if $|V| \to |X|$ maps into $|Z_r|$.
Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. Apply More on Morphisms, Lemma \ref{more-morphisms-lemma-stratify-flat-fp-lqf} to $W \times_X U \to U$. We obtain open subschemes $$ U = U_0 \supset U_1 \supset U_2 \supset \ldots $$ characterized by the property stated in the lemma for the morphism $W \times_X U \to U$. Clearly, the formation of these closed subsets commutes with base change. Setting $R = U \times_X U$ with projection maps $s, t : R \to U$ we conclude that $$ s^{-1}(U_d) = t^{-1}(U_d) $$ as open subschemes of $R$. In other words the open subschemes $U_d \subset U$ are $R$-invariant. This means that $U_d$ is the inverse image of an open subspace $X_d \subset X$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-subspaces-presentation}).
The proof of this lemma is identical to the proof of Lemma \ref{lemma-filter-quasi-compact}. Observe that a quasi-separated space is reasonable, see Lemma \ref{lemma-bounded-fibres} and Definition \ref{definition-very-reasonable}. Hence we find that $U_{n + 1} = \emptyset$ as in Lemma \ref{lemma-filter-reasonable}. At the end of the argument we add that since $X$ is quasi-separated the schemes $U \times_X \ldots \times_X U$ are all quasi-compact. Hence the schemes $W_p$ are quasi-compact. Hence the quotients $V_p = W_p/S_p$ by the symmetric group $S_p$ are quasi-compact schemes.
Assume $E \subset |X|$ is a locally constructible subset of the topological space $|X|$. Let $f : U \to X$ be an \'etale morphism where $U$ is a scheme. We have to show that $f^{-1}(E)$ is locally constructible in $U$. The question is local on $U$ and $X$, hence we may assume that $X$ is quasi-compact, $E \subset |X|$ is constructible, and $U$ is affine. In this case $U \to X$ is quasi-compact, hence $f : |U| \to |X|$ is quasi-compact. Observe that retrocompact opens of $|X|$, resp.\ $U$ are the same thing as quasi-compact opens of $|X|$, resp.\ $U$, see Topology, Lemma \ref{topology-lemma-topology-quasi-separated-scheme}. Thus $f^{-1}(E)$ is constructible by Topology, Lemma \ref{topology-lemma-inverse-images-constructibles}. \medskip\noindent Conversely, assume $E$ is \'etale locally constructible. We want to show that $E$ is locally constructible in the topological space $|X|$. The question is local on $X$, hence we may assume that $X$ is quasi-compact as well as quasi-separated. We will show that in this case $E$ is constructible in $|X|$. Choose open subspaces $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ and surjective \'etale morphisms $f_p : V_p \to U_p$ inducing isomorphisms $f_p^{-1}(T_p) \to T_p = U_p \setminus U_{p + 1}$ where $V_p$ is a quasi-compact separated scheme as in Lemma \ref{lemma-filter-quasi-compact-quasi-separated}. By definition the inverse image $E_p \subset V_p$ of $E$ is locally constructible in $V_p$. Then $E_p$ is constructible in $V_p$ by Properties, Lemma \ref{properties-lemma-constructible-quasi-compact-quasi-separated}. Thus $E_p \cap |f_p^{-1}(T_p)| = E \cap |T_p|$ is constructible in $|T_p|$ by Topology, Lemma \ref{topology-lemma-intersect-constructible-with-closed} (observe that $V_p \setminus f_p^{-1}(T_p)$ is quasi-compact as it is the inverse image of the quasi-compact space $U_{p + 1}$ by the quasi-compact morphism $f_p$). Thus $$ E = (|T_n| \cap E) \cup (|T_{n - 1}| \cap E) \cup \ldots \cup (|T_1| \cap E) $$ is constructible by Topology, Lemma \ref{topology-lemma-collate-constructible-from-constructible}. Here we use that $|T_p|$ is constructible in $|X|$ which is clear from what was said above.
Since both $j$ and $\pi$ are quasi-compact and separated, so is $j \circ \pi$. Let $\nu : Y' \to Y$ be the normalization of $Y$ in $Z$, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-normalization-X-in-Y}. Of course $\nu$ is integral, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-characterize-normalization}. The final statement follows formally from Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-properties-normalization} and \ref{spaces-morphisms-lemma-normalization-in-integral}.