Topology generated by a basis

1. Proposition / Definition

Let \(X\) be a set and \(\mathcal{B}\) a basis. The topology \(\mathcal{T}\) generated by \(\mathcal{B}\) is defined as:

\begin{align*} \mathcal{T} := \left \{\bigcup B_i \vert B_i \in \mathcal{B} \right \} \cup \{\emptyset\} \end{align*}

\(\emptyset \in \mathcal{T}\) is true by definition of the topology; \(X \in \mathcal{T}\) is true by definition of a basis.

is true by definition

Follows from induction, since the second condition of a basis resembles \(\bigcap_{i = 1}^2 B_i\)