1.1 Basics of Locales
For every continuous Function \(f : X \rightarrow Y\) between topological Spaces, there exists a pair of functors \((f^*,f_*)\).
\begin{gather*} f* = f^{-1} : O(Y) \rightarrow O(X)\\ f_* : O(X) \rightarrow O(Y) := A \mapsto \bigcup _{f^*(v) \le A} v\\ \end{gather*}
\(f^*\) is the right adjoint to \(f_*\)
Proof