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

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Theorem

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.

Let $\map \Omega f : \map \Omega {T_2} \to \map \Omega {T_1}$ be the frame homomorphism of $f$


Then:

$\map \Omega f : \map \Omega {T_2} \to \map \Omega {T_1}$ is a frame homomorphism


Proof

Recall the definition of continuous mapping:

$f$ is continuous if and only if $U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$


By definition of frame homomorphism of $f$:

$\map \Omega f$ is the inverse image mapping restricted to $\tau_2 \times \tau_1$

By definition of inverse image mapping:

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


From Inverse Image Mapping is Mapping:

$\map \Omega f : \tau_2 \to \tau_1$ is a well-defined mapping.


$\map \Omega f$ Preserves All Suprema

Let $\TT \subseteq \tau_2$.

By Open Set Axiom $\paren {\text O 1 }$: Union of Open Sets:

$\bigcup \TT \in \tau_2$
$\bigcup \set{\map {\map \Omega f} U : U \in \TT} \in \tau_1$


We have

\(\ds \map {\map \Omega f} {\bigcup \TT}\) \(=\) \(\ds f^{-1} \sqbrk {\bigcup \TT}\) Definition of Frame Homomorphism of Continuous Mapping
\(\ds \) \(=\) \(\ds \bigcup \set{f^{-1} \sqbrk U : U \in \TT}\) Preimage of Union under Mapping
\(\ds \) \(=\) \(\ds \bigcup \set{\map {\map \Omega f} U : U \in \TT}\) Definition of Frame Homomorphism of Continuous Mapping


Hence $\map \Omega f$ is arbitrary join preserving.

$\Box$


$\map \Omega f$ Preserves Finite Infima

Let $\FF \subseteq \tau_2$ be finite.

By Open Set Axiom $\paren {\text O 2 }$: Pairwise Intersection of Open Sets:

$\bigcap \FF \in \tau_2$
$\bigcap \set{\map {\map \Omega f} U : U \in \FF} \in \tau_1$


We have

\(\ds \map {\map \Omega f} {\bigcap \FF}\) \(=\) \(\ds f^{-1} \sqbrk {\bigcap \FF}\) Definition of Frame Homomorphism of Continuous Mapping
\(\ds \) \(=\) \(\ds \bigcap \set{f^{-1} \sqbrk U : U \in \FF}\) Preimage of Intersection under Mapping
\(\ds \) \(=\) \(\ds \bigcap \set{\map {\map \Omega f} U : U \in \FF}\) Definition of Frame Homomorphism of Continuous Mapping


Hence $\map \Omega f$ is finite meet preserving.

$\Box$


It follows that $\map \Omega f$ is a frame homomorphism by definition.

$\blacksquare$


Sources

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