Definition 65.9.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. We define the *dimension of $X$ at $x$* to be the element $\dim _ x(X) \in \{ 0, 1, 2, \ldots , \infty \} $ such that $\dim _ x(X) = \dim _ u(U)$ for any (equivalently some) pair $(a : U \to X, u)$ consisting of an étale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$. See Definition 65.7.5, Lemma 65.7.4, and Descent, Lemma 35.21.2.

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