User:Leigh.Samphier/CategoryTheory/Frame of Open Sets Functor is Contravariant

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Theorem

Let $\mathbf{Top}$ denote the category of topological spaces.

Let $\mathbf{Frm}$ denote the category of frames.


Then:

the open sets functor $\mathbf \Omega : \mathbf{Top} \to \mathbf{Frm}$ is a contravariant functor

Proof

Recall the open sets functor $\mathbf \Omega : \mathbf{Top} \to \mathbf{Frm}$ is defined by:

Object functor:    $\map \Omega T := $ the frame of topological space $T$      
Morphism functor:    $\map \Omega f := $ the frame homomorphism of continuous mapping $f$      


Object Functor is Well-Defined

From User:Leigh.Samphier/Topology/Frame of Topological Space is Frame:

given a topological space $T$:
$\map \Omega T$ is a frame.


Hence the object functor is well-defined.

$\Box$

Arrow Functor is Well-Defined

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

given a continuous mapping $f : T_1 \to T_2$ between topological spaces $T_1$ and $T_2$:
$\map \Omega f : \map \Omega T_2 \to \map \Omega T_1$ is a frame homomorphism.


Hence the arrow functor is well-defined.

$\Box$

Arrow Functor Preserves Composition

Let $T_1 = \struct{S_1, \tau_1}, T_2 = \struct{S_2, \tau_2}, T_3 = \struct{S_3, \tau_3}$ be topological spaces.

Let $f : T_1 \to T_2$ and $g : T_2 \to T_3$ be continuous mappings.


We have

\(\ds \forall U \in \tau_3: \, \) \(\ds \map {\map \Omega {g \circ f} } U\) \(=\) \(\ds \paren{g \circ f}^{-1} \sqbrk U\) Definition of Frame Homomorphism of Continuous Mapping
\(\ds \) \(=\) \(\ds f^{-1} \sqbrk {g^{-1} \sqbrk U}\) Preimage of Subset under Composite Mapping
\(\ds \) \(=\) \(\ds f^{-1} \sqbrk {\map {\map \Omega g} U}\) Definition of Frame Homomorphism of Continuous Mapping
\(\ds \) \(=\) \(\ds \map {\map \Omega f} {\map {\map \Omega g} U}\) Definition of Frame Homomorphism of Continuous Mapping
\(\ds \) \(=\) \(\ds \map {\paren{ {\map \Omega f} \circ {\map \Omega g} } } U\) Definition of Composite Mapping


It follows that:

$\forall f, g : \map \Omega {g \circ f} = {\map \Omega f} \circ {\map \Omega g}$

$\Box$

Arrow Functor Preserves Identities

Let $T = \struct{S, \tau}$ be a topological space.


By definition of frame of topological space:

$\map \Omega T = \struct{\tau, \subseteq}$


We have

\(\ds \forall U \in \tau: \, \) \(\ds \map {\map \Omega {\operatorname{id}_T} } U\) \(=\) \(\ds \map {\map \Omega {\operatorname{id}_S} } U\) Definition of Identity Mapping in Category of Topological Spaces $\mathbf {Top}$
\(\ds \) \(=\) \(\ds \operatorname{id}_S^{-1} \sqbrk U\) Definition of Frame Homomorphism of Continuous Mapping
\(\ds \) \(=\) \(\ds U\) User:Leigh.Samphier/CategoryTheory/Preimage of Subset under Identity Mapping
\(\ds \) \(=\) \(\ds \map {\operatorname{id}_\tau} U\) Definition of Identity Mapping
\(\ds \) \(=\) \(\ds \map {\operatorname{id}_{\map \Omega T} } U\) Definition of Identity Mapping in category of frames $\mathbf {Frm}$


It follows that:

$\forall T : \map \Omega {\operatorname{id}_T } = \operatorname{id}_{\map \Omega T}$

$\Box$


It follows that $\Omega$ is a contravariant functor by definition.

$\blacksquare$

Sources

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