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. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A2Q. Beware of the difference between the letter 'O' and the digit '0'.