Example 65.14.1. Let $k$ be a field of characteristic $\not= 2$. Let $U = \mathbf{A}^1_ k$. Set

where $\Delta = \{ (x, x) \mid x \in \mathbf{A}^1_ k\} $ and $\Gamma = \{ (x, -x) \mid x \in \mathbf{A}^1_ k, x \not= 0\} $. It is clear that $s, t : R \to U$ are étale, and hence $j$ is an étale equivalence relation. The quotient $X = U/R$ is an algebraic space by Theorem 65.10.5. Since $R$ is quasi-compact we see that $X$ is quasi-separated. On the other hand, $X$ is not locally separated because the morphism $j$ is not an immersion.

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