The Stacks project

Lemma 5.5.2. Let $X$ be a set and let $\mathcal{B}$ be a collection of subsets. Assume that $X = \bigcup _{B \in \mathcal{B}} B$ and that given $x \in B_1 \cap B_2$ with $B_1, B_2 \in \mathcal{B}$ there is a $B_3 \in \mathcal{B}$ with $x \in B_3 \subset B_1 \cap B_2$. Then there is a unique topology on $X$ such that $\mathcal{B}$ is a basis for this topology.

Proof. Let $\sigma (\mathcal B)$ be the set of subsets of $X$ which can be written as unions of elements of $\mathcal B$. We claim $\sigma (\mathcal{B})$ is a topology. Namely, the empty set is an element of $\sigma (\mathcal{B})$ (as an empty union) and $X$ is an element of $\sigma (\mathcal{B})$ (as the union of all elements of $\mathcal{B}$). It is clear that $\sigma (\mathcal{B})$ is preserved under unions. Finally, if $U, V \in \sigma (\mathcal{B})$ then write $U = \bigcup _{i \in I} U_ i$ and $V = \bigcup _{j \in J} V_ j$ with $U_ i, V_ j \in \mathcal{B}$ for all $i \in I$ and $j \in J$. Then

\[ U \cap V = \bigcup \nolimits _{i \in I, j \in J} U_ i \cap V_ j \]

The assumption in the lemma tells us that $U_ i \cap V_ j \in \sigma (\mathcal{B})$ hence we see that $U \cap V$ is too. Thus $\sigma (\mathcal{B}$ is a topology. Properties (1) and (2) of Definition 5.5.1 are immediate for this topology. To prove the uniqueness of this topology let $\mathcal T$ be a topology on $X$ such that $\mathcal B$ is a base for it. Then of course every element of $\mathcal{B}$ is in $\mathcal{T}$ by (1) of Definition 5.5.1 and hence $\sigma (\mathcal{B} \subset \mathcal{T}$. Conversely, part (2) of Definition 5.5.1 tells us that every element of $\mathcal{T}$ is a union of elements of $\mathcal{B}$, i.e., $\mathcal{T} \subset \sigma (\mathcal{B})$. This finishes the proof. $\square$

Comments (0)

There are also:

  • 6 comment(s) on Section 5.5: Bases

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Your email address will not be published.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 0D5P. Beware of the difference between the letter 'O' and the digit '0'.