User:Leigh.Samphier/Topology/Definition:Frame Homomorphism of Continuous Mapping

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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 $\map \Omega {T_1} = \struct{\tau_1, \subseteq}$ and $\map \Omega {T_2} = \struct{\tau_2, \subseteq}$ denote the frames of $T_1$ and $T_2$ respectively.


The frame homomorphism of $f$, denoted $\map \Omega f : \map \Omega {T_2} \to \map \Omega {T_1}$, is the inverse image mapping $f^\gets : \powerset {S_2} \to \powerset{S_1}$ restricted to $\tau_2 \times \tau_1$.


That is, the frame homomorphism of $f$ is the mapping $\map \Omega f : \tau_2 \to \tau_1$ defined by:

$\forall U \in \tau_2 : \map {\map \Omega f} U = f^{-1} \sqbrk U$


Also see


Sources

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