8.3 Generated topologies[2BJ]

E251

[0J1] Prerequisites:3.Let \(X\) be a set and \({\mathcal V}⊆ {\mathcal P}(X)\) a family of parts of \(X\); we define \(𝜏\) as the intersection of all topologies that contain \(\mathcal V\) i.e.

\[ 𝜏{\stackrel{.}{=}}\underline⋂\{ 𝜎, 𝜎⊇ \mathcal V, 𝜎 \text{~ topology in ~ }X\} \]

Show that \(𝜏\) is a topology.

\(𝜏\) is the ”topology generated by \(\mathcal V\)”; it is also called ”the smallest topology that contains \(\mathcal V\)”.

[UNACCESSIBLE UUID ’0J2’]

See also the exercises 2.