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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