Lemma 5.14.6. Let \mathcal{I} be a cofiltered category and let i \mapsto X_ i be a diagram over \mathcal{I} in the category of topological spaces. If each X_ i is quasi-compact, Hausdorff, and nonempty, then \mathop{\mathrm{lim}}\nolimits X_ i is nonempty.
Proof. In the proof of Lemma 5.14.5 we have seen that X = \mathop{\mathrm{lim}}\nolimits X_ i is the intersection of the closed subsets
inside the quasi-compact space \prod X_ i where \varphi : j \to k is a morphism of \mathcal{I} and \Gamma _\varphi \subset X_ j \times X_ k is the graph of the corresponding morphism X_ j \to X_ k. Hence by Lemma 5.12.6 it suffices to show any finite intersection of these subsets is nonempty. Assume \varphi _ t : j_ t \to k_ t, t = 1, \ldots , n is a finite collection of morphisms of \mathcal{I}. Since \mathcal{I} is cofiltered, we can pick an object j and a morphism \psi _ t : j \to j_ t for each t. For each pair t, t' such that either (a) j_ t = j_{t'}, or (b) j_ t = k_{t'}, or (c) k_ t = k_{t'} we obtain two morphisms j \to l with l = j_ t in case (a), (b) or l = k_ t in case (c). Because \mathcal{I} is cofiltered and since there are finitely many pairs (t, t') we may choose a map j' \to j which equalizes these two morphisms for all such pairs (t, t'). Pick an element x \in X_{j'} and for each t let x_{j_ t}, resp. x_{k_ t} be the image of x under the morphism X_{j'} \to X_ j \to X_{j_ t}, resp. X_{j'} \to X_ j \to X_{j_ t} \to X_{k_ t}. For any index l \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) which is not equal to j_ t or k_ t for some t we pick an arbitrary element x_ l \in X_ l (using the axiom of choice). Then (x_ i)_{i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})} is in the intersection
by construction and the proof is complete. \square
Comments (0)