Example 66.53.3. This is a continuation of Remark 66.19.4. Consider the algebraic space $X = \mathbf{A}^1_ k/\{ x \sim -x \mid x \not= 0\}$. There are morphisms

$\mathbf{A}^1_ k \longrightarrow X \longrightarrow \mathbf{A}^1_ k$

such that the first arrow is étale surjective, the second arrow is universally injective, and the composition is the map $x \mapsto x^2$. Hence the composition is universally closed. Thus it follows that the map $X \to \mathbf{A}^1_ k$ is a universal homeomorphism, but $X \to \mathbf{A}^1_ k$ is not separated.

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