Definition:Localic Mapping Induced by Continuous Mapping

Definition

Let $T_1 = \struct{S_1, \tau_1}, T_2 = \struct{S_2, \tau_2}$ be topological spaces.

Let $f: T_1 \to \T_2$ be a continuous mapping.

Let $\map \Omega {T_1}, \map \Omega {T_2}$ be the locales of $T_1$ and $T_2$ respectively.

Let $\map \Omega f : \map \Omega {T_2} \to \map \Omega {T_1}$ be the frame homomorphism of $f$ (where $\map \Omega {T_1}, \map \Omega {T_2}$ are considered to be frames).

Then the localic mapping $\upperadjoint {\map \Omega f } : \map \Omega {T_1} \to \map \Omega {T_2}$ that is the upper adjoint of $\map \Omega f$ is called the localic mapping induced by $f$.

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $\text{II}$: Frames and Locales. Spectra, $\S 2.$ Locales and localic maps, $2.4$ Definition