Contents

# Contents

## Definition

The forgetful functor/full and faithful subcategory embedding from compact Hausdorff topological spaces into all topological spaces

$U \;\colon\; Top_{CHaus} \hookrightarrow Top$

$\beta \;\colon\; Top \longrightarrow Top_{CHaus}$

which sends a general topological space to a compact Hausdorff topological space, called its Stone-Čech compactification. This hence exhibits $Top_{CHaus}$ as a reflective subcategory of all of $Top$.

The Stone-Cech compactification is in general “very large”, even for “ordinary” non-compact spaces such as the real line.

For more details see at compactum – Stone-Čech compactification

## Properties

###### Proposition

The unit

$X \longrightarrow \beta X$

of the compactification adjunction $(\beta \dashv U)$ is an embedding precisely for $X$ a Tychonoff space.

## Examples

###### Example

The Stone-Cech compactification of a discrete topological space is an extremally disconnected topological space. By a theorem by Gleason, these are precisely the projective objects in the category of compact Hausdorff topological spaces.

Such spaces appear for instance as connected components of w-contractible rings as objects in the pro-étale site. See (Bhatt-Scholze 13).

## References

Lecture notes include

Discussion in the context of the pro-etale site is in