User:Leigh.Samphier/Topology/Order Isomorphism is Isomorphism in Loc*

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Theorem

Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.

Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be locales.

Let $f:S_1 \to S_2$ be a localic mapping of $\mathbf{Loc_*}$.

Then:

$f$ is an isomorphism of $\mathbf{Loc_*}$ if and only if $f$ is an order isomorphism.

Proof

$\blacksquare$