Lemma 70.12.3. Let $k$ be a field. Let $X$, $Y$ be algebraic spaces over $k$. Assume $X$ is geometrically connected over $k$. Then the projection morphism

$p : X \times _ k Y \longrightarrow Y$

induces a bijection between connected components.

Proof. Let $y \in |Y|$ be represented by a morphism $\mathop{\mathrm{Spec}}(K) \to Y$ be a morphism where $K$ is a field. The fibre of $|X \times _ k Y| \to |Y|$ over $y$ is the image of $|Y_ K| \to |X \times _ k Y|$ by Properties of Spaces, Lemma 64.4.3. Thus these fibres are connected by our assumption that $Y$ is geometrically connected. By Morphisms of Spaces, Lemma 65.6.6 the map $|p|$ is open. Thus we may apply Topology, Lemma 5.7.6 to conclude. $\square$

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