User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification

Definition

Locale

Let $\mathbf{Loc}$ denote the category of locales.

Let $\mathbf{KCRegLoc}$ denote the category of compact completely regular locales.

Let $i: \mathbf{KCRegLoc} \to \mathbf{Loc}$ be the inclusion functor.

Let $\beta: \mathbf{Loc} \to \mathbf{KCRegLoc}$ be a left adjoint functor of $i$.

For any $A \in \mathbf{Loc}$:

$\beta A$ is called a Stone-Čech Compactification of $A$.

Topological Space

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

Let $\mathbf{KHausTop}$ denote the category of compact Hausdorff spaces.

Let $i: \mathbf{KHausTop} \to \mathbf{Top}$ be the inclusion functor.

Let $\beta: \mathbf{Top} \to \mathbf{KHausTop}$ be a left adjoint functor of $i$.

For any $A \in \mathbf{Top}$:

$\beta A$ is called a Stone-Čech Compactification of $A$.