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