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 $\tuple{\beta, \iota, \alpha}$ be an adjunction of $\mathbf{KHausTop}$ and $\mathbf{Top}$ where:

$\mathbf{\mathbf{Top}} \to \mathbf{\mathbf{KHausTop}}$ denotes a functor

$\iota: \mathbf{\mathbf{KHausTop}} \to \mathbf{\mathbf{Top}}$ denotes the inclusion functor

Let $\eta:\operatorname{id}_{\mathbf{\mathbf{Top}}} \to \iota \beta$ be the unit of the adjunction $\tuple{\beta, \iota, \alpha}$ where:

$\operatorname{id}_{\mathbf{\mathbf{Top}}}$ denotes the identity functor on $\mathbf{Top}$

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

$\eta_A: A \to \beta A$ 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 up to Isomorphism

Sources

Johnstone