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

Definition

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 $A$.

Also see

• Inclusion from Compact Hasdorff Spaces into Topological Spaces has Left Adjoint

• Stone-Čech Compactifications of Topological Spaces are Unique upto Isomorphism

Sources

Johnstone