hcju/stacksqa huggingface dataset:
Choose an affine scheme $V$ with a point $z' \in V$ and an \'etale morphism $V \to Y$ mapping $z'$ to $z$. Let $Z' \subset V$ be the scheme theoretic image of $X \times_Y V \to V$. By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} we have $Z' = Z \times_Y V$. Thus $z' \in Z'$. Since $f$ is quasi-compact and $V$ is affine we see that $X \times_Y V$ is quasi-compact. Hence there exists an affine scheme $W$ and a surjective \'etale morphism $W \to X \times_Y V$. Then $Z' \subset V$ is also the scheme theoretic image of $W \to V$. By Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image} we can choose a diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & W \ar[r] \ar[d] & X \times_Y V \ar[d] \ar[r] & X \ar[d] \\ \Spec(A) \ar[r] & Z' \ar[r] & V \ar[r] & Y } $$ such that the closed point of $\Spec(A)$ maps to $z'$. Composing with $Z' \to Z$ and $W \to X \times_Y V \to X$ we obtain a solution.
By Lemma \ref{lemma-quasi-compact-permanence} the morphism $s : V \to X$ is quasi-compact. Hence the construction of the scheme theoretic image $Y'$ of $s$ commutes with restriction to opens by Lemma \ref{lemma-quasi-compact-scheme-theoretic-image}. In particular, we see that $Y' \cap f^{-1}(V)$ is the scheme theoretic image of a section of the separated morphism $f^{-1}(V) \to V$. Since a section of a separated morphism is a closed immersion (Lemma \ref{lemma-section-immersion}), we conclude that $Y' \cap f^{-1}(V) \to V$ is an isomorphism as desired.
We will use the characterization of Morphisms, Lemma \ref{morphisms-lemma-characterize-scheme-theoretically-dense}. Assume $V \subset X$ is an open and $g \in \Gamma(V, \mathcal{O}_V)$ is a function which restricts to zero on $f^{-1}(W) \cap V$. We have to show that $g = 0$. Assume $g \not = 0$ to get a contradiction. By More on Morphisms, Lemma \ref{more-morphisms-lemma-go-down-with-annihilators} we may shrink $V$, find an open $U \subset S$ fitting into a commutative diagram $$ \xymatrix{ V \ar[r] \ar[d]_\pi & X \ar[d]^f \\ U \ar[r] & S, } $$ a quasi-coherent subsheaf $\mathcal{F} \subset \mathcal{O}_U$, an integer $r > 0$, and an injective $\mathcal{O}_U$-module map $\mathcal{F}^{\oplus r} \to \pi_*\mathcal{O}_V$ whose image contains $g|_V$. Say $(g_1, \ldots, g_r) \in \Gamma(U, \mathcal{F}^{\oplus r})$ maps to $g$. Then we see that $g_i|_{W \cap U} = 0$ because $g|_{f^{-1}W \cap V} = 0$. Hence $g_i = 0$ because $\mathcal{F} \subset \mathcal{O}_U$ and $W$ is scheme theoretically dense in $S$. This implies $g = 0$ which is the desired contradiction.
Observe that if $V \to V'$ is a morphism of algebraic spaces \'etale over $X$, and $Z \subset V$, resp.\ $Z' \subset V'$ is the scheme theoretic closure of $U \times_X V$, resp.\ $U \times_X V'$ in $V$, resp.\ $V'$, then $Z$ maps into $Z'$. Thus if $V \to V'$ is surjective and \'etale then $Z = V$ implies $Z' = V'$. Next, note that an \'etale morphism is flat, locally of finite presentation, and locally quasi-finite (see Morphisms, Section \ref{morphisms-section-etale}). Thus Lemma \ref{lemma-scheme-theoretically-dense-representable} implies that if $V$ and $V'$ are schemes, then $Z' = V'$ implies $Z = V$. A formal argument using that every algebraic space has an \'etale covering by a scheme shows that (1) and (2) are equivalent.
If $\mathcal{O}_X \to j_*\mathcal{O}_U$ is injective, then the same is true when restricted to any algebraic space $V$ \'etale over $X$. Hence the scheme theoretic closure of $U \times_X V$ in $V$ is equal to $V$, see proof of Lemma \ref{lemma-scheme-theoretic-image}. Conversely, assume the scheme theoretic closure of $U \times_X V$ is equal to $V$ for all $V$ \'etale over $X$. Suppose that $\mathcal{O}_X \to j_*\mathcal{O}_U$ is not injective. Then we can find an affine, say $V = \Spec(A)$, \'etale over $X$ and a nonzero element $f \in A$ such that $f$ maps to zero in $\Gamma(V \times_X U, \mathcal{O})$. In this case the scheme theoretic closure of $V \times_X U$ in $V$ is clearly contained in $\Spec(A/(f))$ a contradiction.
Let $W \to X$ be any \'etale morphism. Consider the map $\mathcal{O}(W) \to \mathcal{O}(W \times_X V) \to \mathcal{O}(W \times_X (V \cap U))$. By Lemma \ref{lemma-characterize-scheme-theoretically-dense} both maps are injective. Hence the composite is injective. Hence by Lemma \ref{lemma-characterize-scheme-theoretically-dense} $U \cap V$ is scheme theoretically dense in $X$.
We are going to use Morphisms, Lemma \ref{morphisms-lemma-universally-injective} without further mention. Suppose that $f$ is universally injective. Then for any field $K$ and any morphism $\Spec(K) \to Y$ the morphism of schemes $\Spec(K) \times_Y X \to \Spec(K)$ is universally injective. Hence there exists at most one section of the morphism $\Spec(K) \times_Y X \to \Spec(K)$. Hence the map $X(K) \to Y(K)$ is injective. Conversely, suppose that for every field $K$ the map $X(K) \to Y(K)$ is injective. Let $T \to Y$ be a morphism from a scheme into $Y$, and consider the base change $f_T : T \times_Y X \to T$. For any field $K$ we have $$ (T \times_Y X)(K) = T(K) \times_{Y(K)} X(K) $$ by definition of the fibre product, and hence the injectivity of $X(K) \to Y(K)$ guarantees the injectivity of $(T \times_Y X)(K) \to T(K)$ which means that $f_T$ is universally injective as desired.
Assume (1). Let $g : Y' \to Y$ be a morphism of algebraic spaces, and denote $f' : Y' \times_Y X \to Y'$ the base change of $f$. Let $K_i$, $i = 1, 2$ be fields and let $\varphi_i : \Spec(K_i) \to Y' \times_Y X$ be morphisms such that $f' \circ \varphi_1$ and $f' \circ \varphi_2$ define the same element of $|Y'|$. By definition this means there exists a field $\Omega$ and embeddings $\alpha_i : K_i \subset \Omega$ such that the two morphisms $f' \circ \varphi_i \circ \alpha_i : \Spec(\Omega) \to Y'$ are equal. Here is the corresponding commutative diagram $$ \xymatrix{ \Spec(\Omega) \ar@/_5ex/[ddrr] \ar[rd]^{\alpha_1} \ar[r]_{\alpha_2} & \Spec(K_2) \ar[rd]^{\varphi_2} \\ & \Spec(K_1) \ar[r]^{\varphi_1} & Y' \times_Y X  \ar[d]^{f'} \ar[r]^{g'} &  X \ar[d]^f \\ & & Y' \ar[r]^g & Y. } $$ In particular the compositions $g \circ f' \circ \varphi_i \circ \alpha_i$ are equal. By assumption (1) this implies that the morphism $g' \circ \varphi_i \circ \alpha_i$ are equal, where $g' : Y' \times_Y X \to X$ is the projection. By the universal property of the fibre product we conclude that the morphisms $\varphi_i \circ \alpha_i : \Spec(\Omega) \to Y' \times_Y X$ are equal. In other words $\varphi_1$ and $\varphi_2$ define the same point of $Y' \times_Y X$. We conclude that (2) holds. \medskip\noindent Assume (2). Let $K$ be a field over $S$, and let $a, b : \Spec(K) \to X$ be two morphisms such that $f \circ a = f \circ b$. Denote $c : \Spec(K) \to Y$ the common value. By assumption $|\Spec(K) \times_{c, Y} X| \to |\Spec(K)|$ is injective. This means there exists a field $\Omega$ and embeddings $\alpha_i : K \to \Omega$ such that $$ \xymatrix{ \Spec(\Omega) \ar[r]_{\alpha_1} \ar[d]_{\alpha_2} & \Spec(K) \ar[d]^a \\ \Spec(K) \ar[r]^-b & \Spec(K) \times_{c, Y} X } $$ is commutative. Composing with the projection to $\Spec(K)$ we see that $\alpha_1 = \alpha_2$. Denote the common value $\alpha$. Then we see that $\{\alpha : \Spec(\Omega) \to \Spec(K)\}$ is a fpqc covering of $\Spec(K)$ such that the two morphisms $a, b$ become equal on the members of the covering. By Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc} we conclude that $a = b$. We conclude that (1) holds. \medskip\noindent Assume (3). Let $x, x' \in |X|$ be a pair of points such that $f(x) = f(x')$ in $|Y|$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} we see there exists a $x'' \in |X \times_Y X|$ whose projections are $x$ and $x'$. By assumption and Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-surjective} there exists a $x''' \in |X|$ with $\Delta_{X/Y}(x''') = x''$. Thus $x = x'$. In other words $f$ is injective. Since condition (3) is stable under base change we see that $f$ satisfies (2). \medskip\noindent Assume (2). Then in particular $|X \times_Y X| \to |X|$ is injective which implies immediately that $|\Delta_{X/Y}| : |X| \to |X \times_Y X|$ is surjective, which implies that $\Delta_{X/Y}$ is surjective by Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-surjective}.
We will use that being universally injective is preserved under base change (Lemma \ref{lemma-base-change-universally-injective}) without further mention in this proof. It is clear that (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4). \medskip\noindent Assume $g : Z \to Y$ as in (4). Let $y : \Spec(K) \to Y$ be a morphism from the spectrum of a field into $Y$. By assumption we can find an extension field $\alpha : K \subset K'$ and a morphism $z : \Spec(K') \to Z$ such that $y \circ \alpha = g \circ z$ (with obvious abuse of notation). By assumption the morphism $Z \times_Y X \to Z$ is universally injective, hence there is at most one lift of $g \circ z : \Spec(K') \to Y$ to a morphism into $X$. Since $\{\alpha : \Spec(K') \to \Spec(K)\}$ is a fpqc covering this implies there is at most one lift of $y : \Spec(K) \to Y$ to a morphism into $X$, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-sheaf-fpqc}. Thus we see that (1) holds. \medskip\noindent We omit the verification that (5) is equivalent to (1).
It is clear that (1) implies (2), that (2) implies (3), and that (3) implies (4) by taking $V$ to be a disjoint union of affines \'etale over $Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. Assume $V \to Y$ is as in (4). Then for every affine open $W$ of $V$ we see that $W \times_Y X$ is an affine open of $V \times_Y X$. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} we conclude that $V \times_Y X$ is a scheme. Moreover the morphism $V \times_Y X \to V$ is affine. This means we can apply Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} because the class of affine morphisms satisfies all the required properties (see Morphisms, Lemmas \ref{morphisms-lemma-base-change-affine} and Descent, Lemmas \ref{descent-lemma-descending-property-affine} and \ref{descent-lemma-affine}). The conclusion of applying this lemma is that $f$ is representable and affine, i.e., (1) holds. \medskip\noindent The equivalence of (1) and (5) follows from the fact that being affine is Zariski local on the target (the reference above shows that being affine is in fact fpqc local on the target).
This lemma is the analogue of Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-algebras}. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. We will construct an affine morphism of algebraic spaces $\pi : Y = \underline{\Spec}_X(\mathcal{A}) \to X$ with $\pi_*\mathcal{O}_Y \cong \mathcal{A}$. To do this, choose a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$. As usual denote $R = U \times_X U$ with projections $s, t : R \to U$. Denote $\psi : R \to X$ the composition $\psi = \varphi \circ s = \varphi \circ t$. By the aforementioned lemma there exists an affine morphisms of schemes $\pi_0 : V \to U$ and $\pi_1 : W \to R$ with $\pi_{0, *}\mathcal{O}_V \cong \varphi^*\mathcal{A}$ and $\pi_{1, *}\mathcal{O}_W \cong \psi^*\mathcal{A}$. Since the construction is compatible with base change there exist morphisms $s', t' : W \to V$ such that the diagrams $$ \vcenter{ \xymatrix{ W \ar[r]_{s'} \ar[d] & V \ar[d] \\ R \ar[r]^s & U } } \quad\text{and}\quad \vcenter{ \xymatrix{ W \ar[r]_{t'} \ar[d] & V \ar[d] \\ R \ar[r]^t & U } } $$ are cartesian. It follows that $s', t'$ are \'etale. It is a formal consequence of the above that $(t', s') : W \to V \times_S V$ is a monomorphism. We omit the verification that $W \to V \times_S V$ is an equivalence relation (hint: think about the pullback of $\mathcal{A}$ to $U \times_X U \times_X U = R \times_{s, U, t} R$). The quotient sheaf $Y = V/W$ is an algebraic space, see Spaces, Theorem \ref{spaces-theorem-presentation}. By Groupoids, Lemma \ref{groupoids-lemma-criterion-fibre-product} we see that $Y \times_X U \cong V$. Hence $Y \to X$ is affine by Lemma \ref{lemma-affine-local}. Finally, the isomorphism of $$ (Y \times_X U \to U)_*\mathcal{O}_{Y \times_X U} = \pi_{0, *}\mathcal{O}_V \cong \varphi^*\mathcal{A} $$ is compatible with glueing isomorphisms, whence $(Y \to X)_*\mathcal{O}_Y \cong \mathcal{A}$ by Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent}. We omit the verification that this construction is compatible with base change.
Proof of (1). The base change $X \times_B Y \to Y$ is affine by Lemma \ref{lemma-base-change-affine}. The morphism $(1, g) : X \to X \times_B Y$ is the base change of $Y \to Y \times_B Y$ by the morphism $X \times_B Y \to Y \times_B Y$. Hence it is affine by Lemma \ref{lemma-base-change-affine}. The composition of affine morphisms is affine (see Lemma \ref{lemma-composition-affine}) and (1) follows. Part (2) follows from (1) as a closed immersion is affine (see Lemma \ref{lemma-closed-immersion-affine}) and $Y/B$ separated means $\Delta$ is a closed immersion. Parts (3) and (4) are special cases of (1) and (2).
Let $U \to X$ be an \'etale morphism with $U$ affine. To prove the lemma we have to show that $\Spec(A) \times_X U$ is affine, see Lemma \ref{lemma-affine-local}. Since $X$ is quasi-separated the scheme $\Spec(A) \times_X U$ is quasi-compact. Moreover, the projection morphism $\Spec(A) \times_X U \to \Spec(A)$ is \'etale. Hence this morphism has finite discrete fibers and moreover the topology on $\Spec(A)$ is discrete. Thus $\Spec(A) \times_X U$ is a scheme whose underlying topological space is a finite discrete set. We are done by Schemes, Lemma \ref{schemes-lemma-scheme-finite-discrete-affine}.
It is clear that (1) implies (2) and that (2) implies (3) by taking $V$ to be a disjoint union of affines \'etale over $Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. Assume $V \to Y$ is as in (3). Then for every affine open $W$ of $V$ we see that $W \times_Y X$ is a quasi-affine open of $V \times_Y X$. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} we conclude that $V \times_Y X$ is a scheme. Moreover the morphism $V \times_Y X \to V$ is quasi-affine. This means we can apply Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale} because the class of quasi-affine morphisms satisfies all the required properties (see Morphisms, Lemmas \ref{morphisms-lemma-base-change-quasi-affine} and Descent, Lemmas \ref{descent-lemma-descending-property-quasi-affine} and \ref{descent-lemma-quasi-affine}). The conclusion of applying this lemma is that $f$ is representable and quasi-affine, i.e., (1) holds. \medskip\noindent The equivalence of (1) and (4) follows from the fact that being quasi-affine is Zariski local on the target (the reference above shows that being quasi-affine is in fact fpqc local on the target).
Let $U \to X$ be a surjective morphism where $U$ is a scheme. Since we may check whether $f$ is quasi-affine after base change to $U$ (Lemma \ref{lemma-quasi-affine-local}), since $f_*\mathcal{O}_Y|_V$ is equal to $(Y \times_X U \to U)_*\mathcal{O}_{Y \times_X U}$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}), and since formation of relative spectrum commutes with base change (Lemma \ref{lemma-affine-equivalence-algebras}), we see that the assertion reduces to the case that $X$ is a scheme. If $X$ is a scheme and either $f$ is quasi-affine or $Y \to \underline{\Spec}_X(f_*\mathcal{O}_Y)$ is an open immersion, then $Y$ is a scheme as well. Thus we reduce to Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine}.
Let us prove the equivalence of (1) and (2). The implication (1) $\Rightarrow$ (2) is immediate (taking into account Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}). Assume $$ \xymatrix{ U \ar[d] \ar[r]_h & V \ar[d] \\ X \ar[r]^f & Y } \quad\quad \xymatrix{ U' \ar[d] \ar[r]_{h'} & V' \ar[d] \\ X \ar[r]^f & Y } $$ are two diagrams as in the lemma. Assume $U \to X$ is surjective and $h$ has property $\mathcal{P}$. To show that (2) implies (1) we have to prove that $h'$ has $\mathcal{P}$. To do this consider the diagram $$ \xymatrix{ U \ar[d]_h & U \times_X U' \ar[l] \ar[d]^{(h, h')} \ar[r] & U' \ar[d]^{h'} \\ V & V \times_Y V' \ar[l] \ar[r] & V' } $$ By Descent, Lemma \ref{descent-lemma-local-source-target-characterize} we see that $h$ has $\mathcal{P}$ implies $(h, h')$ has $\mathcal{P}$ and since $U \times_X U' \to U'$ is surjective this implies (by the same lemma) that $h'$ has $\mathcal{P}$. \medskip\noindent If $X$ and $Y$ are representable, then Descent, Lemma \ref{descent-lemma-local-source-target-characterize} applies which shows that (1) and (2) are equivalent to $f$ having $\mathcal{P}$. \medskip\noindent Finally, suppose $f$ is representable, and $U, V, a, b, h$ are as in part (2) of the lemma, and that $\mathcal{P}$ is preserved under arbitrary base change. We have to show that for any scheme $Z$ and morphism $Z \to X$ the base change $Z \times_Y X \to Z$ has property $\mathcal{P}$. Consider the diagram $$ \xymatrix{ Z \times_Y U \ar[d] \ar[r] & Z \times_Y V \ar[d] \\ Z \times_Y X \ar[r] & Z } $$ Note that the top horizontal arrow is a base change of $h$ and hence has property $\mathcal{P}$. The left vertical arrow is \'etale and surjective and the right vertical arrow is \'etale. Thus Descent, Lemma \ref{descent-lemma-local-source-target-characterize} once again kicks in and shows that $Z \times_Y X \to Z$ has property $\mathcal{P}$.
See Descent, Lemma \ref{descent-lemma-local-source-target-global-implies-local} for (1). The implication (1)(a) $\Rightarrow$ (2)(b) follows on letting $X' = a(U) \subset X$ given a diagram as in Lemma \ref{lemma-local-source-target-at-point}. The implication (2)(b) $\Rightarrow$ (1)(a) is clear. The equivalence of (3)(a) and (3)(b) follows from the corresponding result for morphisms of schemes, see Descent, Lemma \ref{descent-lemma-local-source-target-local-implies-global}.
Each of the conditions (2), (3), (4), (5), (6), (7), and (9) imply condition (8) in a straightforward manner. For example, if (5) holds, then we can choose a scheme $V$ which is a disjoint union of affines and a surjective morphism $V \to Y$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}). Then $V \times_Y X \to V$ is locally of finite type by (5). Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Then $U \to V$ is locally of finite type by Lemma \ref{lemma-composition-finite-type}. Hence (8) is true. \medskip\noindent The conditions (1), (7), and (8) are equivalent by definition. \medskip\noindent To finish the proof, we show that (1) implies all of the conditions (2), (3), (4), (5), (6), and (9). For (2) this is immediate. For (3), (4), (5), and (9) this follows from the fact that being locally of finite type is preserved under base change, see Lemma \ref{lemma-base-change-finite-type}. For (6) we can take $U = X$ and we're done.
Let $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ be a commutative diagram where $U$, $V$ are schemes and the vertical arrows are surjective \'etale. If $f$ is locally of finite type, then $U \to V$ is locally of finite type. If $Y$ is locally Noetherian, then $V$ is locally Noetherian. By Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian} we see that $U$ is locally Noetherian, which means that $X$ is locally Noetherian.
We can find a diagram $$ \xymatrix{ U \ar[r] \ar[d] & V \ar[r] \ar[d] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } $$ where $U$, $V$, $W$ are schemes, the vertical arrows are \'etale and surjective, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. At this point we can use Lemma \ref{lemma-finite-type-local} and Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type} to conclude.
Choose a diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$, $V$ schemes over $S$ and vertical arrows surjective and \'etale, see Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. Since $f$ is locally of finite type we see that $U \to V$ is locally of finite type. \medskip\noindent Assume (1) and let $\overline{y} \in Y(k)$. Then $U \to Y$ is surjective and locally of finite type by Lemmas \ref{lemma-composition-surjective} and \ref{lemma-composition-finite-type}. Let $Z = U \times_{Y, \overline{y}} \Spec(k)$. This is a scheme. The projection $Z \to \Spec(k)$ is surjective and locally of finite type by Lemmas \ref{lemma-base-change-surjective} and \ref{lemma-base-change-finite-type}. It follows from Varieties, Lemma \ref{varieties-lemma-locally-finite-type-Jacobson} that $Z$ has a $k$ valued point $\overline{z}$. The image $\overline{x} \in X(k)$ of $\overline{z}$ maps to $\overline{y}$ as desired. \medskip\noindent Assume (2). By Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-surjective} it suffices to show that $|X| \to |Y|$ is surjective. Let $y \in |Y|$. Choose a $u \in U$ mapping to $y$. Let $k \supset \kappa(u)$ be an algebraic closure. Denote $\overline{u} \in U(k)$ the corresponding point and $\overline{y} \in Y(k)$ its image. By assumption there exists a $\overline{x} \in X(k)$ mapping to $\overline{y}$. Then it is clear that the image $x \in |X|$ of $\overline{x}$ maps to $y$.
Let $u \in U$ and $U \to X$ be as in (2). Then $\Spec(\kappa(u)) \to U$ is of finite type, and $U \to X$ is representable and locally of finite type (by the general principle Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication} and Morphisms, Lemmas \ref{morphisms-lemma-etale-locally-finite-presentation} and \ref{morphisms-lemma-finite-presentation-finite-type}). Hence we see (1) holds by Lemma \ref{lemma-composition-finite-type}. \medskip\noindent Conversely, assume $\Spec(k) \to X$ is locally of finite type and represents $x$. Let $U \to X$ be a surjective \'etale morphism where $U$ is a scheme. By assumption $U \times_X \Spec(k) \to U$ is locally of finite type. Pick a finite type point $v$ of $U \times_X \Spec(k)$ (there exists at least one, see Morphisms, Lemma \ref{morphisms-lemma-identify-finite-type-points}). By Morphisms, Lemma \ref{morphisms-lemma-finite-type-points-morphism} the image $u \in U$ of $v$ is a finite type point of $U$. Hence by Morphisms, Lemma \ref{morphisms-lemma-identify-finite-type-points} after shrinking $U$ we may assume that $u$ is a closed point of $U$, i.e., (2) holds.
We have $f(X_{\text{ft-pts}}) \subset Y_{\text{ft-pts}}$ by Lemma \ref{lemma-finite-type-points-morphism}. Let $y \in |Y|$ be a finite type point. Represent $y$ by a morphism $\Spec(k) \to Y$ which is locally of finite type. As $f$ is surjective the algebraic space $X_k = \Spec(k) \times_Y X$ is nonempty, therefore has a finite type point $x \in |X_k|$ by Lemma \ref{lemma-identify-finite-type-points}. Now $X_k \to X$ is a morphism which is locally of finite type as a base change of $\Spec(k) \to Y$ (Lemma \ref{lemma-base-change-finite-type}). Hence the image of $x$ in $X$ is a finite type point by Lemma \ref{lemma-finite-type-points-morphism} which maps to $y$ by construction.
Let $i : Z \to X$ be the reduced induce subspace structure on $T$, see Remark \ref{remark-space-structure-locally-closed-subset}. Any immersion is locally of finite type, see Lemma \ref{lemma-immersion-locally-finite-type}. Hence by Lemma \ref{lemma-finite-type-points-morphism} we see $Z_{\text{ft-pts}} \subset X_{\text{ft-pts}} \cap T$. Finally, any nonempty affine scheme $U$ with an \'etale morphism towards $Z$ has at least one closed point. Hence $Z$ has at least one finite type point by Lemma \ref{lemma-identify-finite-type-points}. The lemma follows.