The Stacks project

Proof. If (2) holds, then $X_{red}$ is a separated algebraic space by applying Properties of Spaces, Lemma 66.3.3 to morphisms $U \to X_{red}$ and $V \to X_{red}$ with $U, V$ affine and using that $U \times _{X_{red}} V = U \times _ X V$.

Assume (1). Let $U \to X$ and $V \to X$ be as in (2). Observe that $U \times _ X V$ is a scheme by Lemma 87.11.2. Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$. Then

is a thickening of schemes. It follows that $(U \times _ X V)_{red} = (U_{red} \times _{X_{red}} V_{red})_{red}$. In particular, we see that $(U \times _ X V)_{red}$ is an affine scheme and that

is surjective, see Properties of Spaces, Lemma 66.3.3. Then $U \times _ X V$ is affine by Limits of Spaces, Proposition 70.15.2. On the other hand, the morphism $U \times _ X V \to U \times V$ of affine schemes is the composition

The first morphism is a monomorphism and locally of finite type (Lemma 87.11.2). The second morphism is an immersion (Schemes, Lemma 26.21.9). Hence the composition is a monomorphism which is locally of finite type. On the other hand, the composition is integral as the map on underlying reduced affine schemes is a closed immersion by the above and hence universally closed (use Morphisms, Lemma 29.44.7). Thus the ring map

is an epimorphism which is integral of finite type hence finite hence surjective (use Morphisms, Lemma 29.44.4 and Algebra, Lemma 10.107.6). $\square$