User:Leigh.Samphier/Topology/Identity Mapping is Localic Mapping

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Theorem

Let $L = \struct{S, \preceq}$ be a locale.

Let $\operatorname{id}_S : L \to L$ be the identity mapping on $S$.

Then:

$\operatorname{id}_S$ is a localic mapping.

Proof

By definition of localic mapping, we need to show that $\operatorname{id}_S$ is the upper adjoint of a Galois connection.

It will be shown that $\tuple{\operatorname{id}_S, \operatorname{id}_S}$ is a Galois connection.

We have

\(\ds \forall x, y \in S: \, \)

\(\ds x \preceq \map {\operatorname{id}_S} y\)

\(\iff\)

\(\ds x \preceq y\)

Definition of Identity Mapping

\(\ds \)

\(\iff\)

\(\ds \map {\operatorname{id}_S} x \preceq y\)

Definition of Identity Mapping

The result follows.

$\blacksquare$

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter II: Frames and Locales. Spectra, $\S 2.2$