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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)