nLab Stone-Čech compactification

Definition

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

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

has a left adjoint

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

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

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

For more details see at compact Hausdorff space – 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-Čech 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 2013.

Related concepts

• compactification

• one-point compactification

• end compactification

• Bohr compactification

• Kaluza-Klein compactification

References

Lecture notes include

• Tarun Chitra: The Stone-Cech Compactification (2009) [pdf]

• Brian Bockelman: The Stone-Cech compactification [archive]

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

• Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes (arXiv:1309.1198)

See also

• Wikipedia, Stone–Čech compactification