Basis (topology)
1. Definition
Let \(X\) be a set and \(\mathcal{B} \subseteq \mathcal{P}(X)\) a collection of subsets. \(\mathcal{B}\) is a basis for a topology on \(X\), if the following statements hold:
1.1. a)
\begin{align*}
\bigcup_{B \in \mathcal{B}} B = X
\end{align*}
1.2. b)
For \(B_1,B_2 \in \mathcal{B}\) and suitable \(B_j\) (possibly \(J = \emptyset\)) it holds that:
\begin{align*} \bigcap_{j \in J} B_j = B_1 \cap B_2 \end{align*}