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The Stacks project

Lemma 5.14.3. Let \mathcal{I} be a cofiltered category. Let i \mapsto X_ i be a diagram of topological spaces over \mathcal{I}. Let X be a topological space such that

  1. X = \mathop{\mathrm{lim}}\nolimits X_ i as a set (denote f_ i the projection maps),

  2. the sets f_ i^{-1}(U_ i) for i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) and U_ i \subset X_ i open form a basis for the topology of X.

Then X is the limit of the X_ i as a topological space.

Proof. Follows from the description of the limit topology in Lemma 5.14.2. \square


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