Lemma 27.2.7. A partition $X = \coprod _{i \in I} X_ i$ of a scheme $X$ with retrocompact parts is locally finite if and only if the parts are locally constructible.

**Proof.**
See Topology, Definitions 5.12.1, 5.28.1, and 5.28.4 for the definitions of retrocompact, partition, and locally finite.

If the partition is locally finite and $U \subset X$ is an affine open, then we see that $U = \coprod _{i \in I} U \cap X_ i$ is a finite partition (more precisely, all but a finite number of its parts are empty). Hence $U \cap X_ i$ is quasi-compact and its complement is retrocompact in $U$ as a finite union of retrocompact parts. Thus $U \cap X_ i$ is constructible by Topology, Lemma 5.15.13. It follows that $X_ i$ is locally constructible by Lemma 27.2.1.

Assume the parts are locally constructible. Then for any affine open $U \subset X$ we obtain a covering $U = \coprod X_ i \cap U$ by constructible subsets. Since the constructible topology is quasi-compact, see Topology, Lemma 5.23.2, this covering has a finite refinement, i.e., the partition is locally finite. $\square$

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