# Is Stone-Čech compactification the only one with universal property?

The Stone-Čech compactification (also called the Čech-Stone compactification) is the "biggest" compactification of a topological space. (I am working with Hausdorff compactifications only, to be clear.)

Formally,

The Stone–Čech compactification of the topological space $$X$$ is a compact Hausdorff space $$βX$$ together with a continuous map $$iX : X → βX$$ that has the following universal property: any continuous map $$f : X → K$$, where K is a compact Hausdorff space, extends uniquely to a continuous map $$βf : βX → K$$, i.e. $$(βf)i_X = f$$.

My questions are: (1) Is Stone-Čech known as the only compactification with the universal property?

(2) Are there any compactification with this property, but without requiring the $$\beta f$$ to be unique?

• From memory, sorry if it's wrong but... (a) The universal property should guarantee uniqueness (up to homeomorphism), proof: suppose two constructions have the universal property, chase diagrams; (b) I can imagine a compactification $X\to\beta X\sqcup\{ *\}$ where we add an additional (isolated) point $*$, this does not have universal property and probably has no particular theoretical significance either!
– user700480
Jan 1, 2022 at 18:51
• @StinkingBishop Thank you! For the (a), I also have a feeling there shouldn´t exist another Stone-Čech-like compactifiaction, but wasnt sure about the proof. Jan 1, 2022 at 19:32
• Do try to chase diagrams, though...
– user700480
Jan 1, 2022 at 19:34

To answer (1), that universal property does, indeed, determine the Stone-Čech compactification, in the same manner as many universal properties, namely: if $$\beta X$$ and $$\beta' X$$ are two compactifications of $$X$$ that satisfy the stated properties then there exists a unique homeomorphism $$F : \beta X \to \beta' X$$ whose restriction to $$X$$ is the identity. The proof of existence and uniqueness of $$F$$ is quite similar to many such "universal property" arguments:
• Apply the existence portion of the universal property for $$\beta' X$$ with $$K=\beta X$$ to construct a continuous map $$F : \beta X \to \beta' X$$ whose restriction to $$X$$ is the identity.
• Apply the existence portion of the universal property for $$\beta X$$ with $$K = \beta' X$$ to construct a continuous map $$F' : \beta' X \to \beta X$$ whose restriction to $$X$$ is the identity.
• Apply the uniqueness portion of the universal property for $$\beta X$$ with $$K=\beta X$$ to conclude that $$F' \circ F : \beta X \to \beta X$$ is the identity map.
• Apply the uniqueness portion of the universal property for $$\beta' X$$ with $$K=\beta' X$$ to conclude that $$F \circ F' : \beta' X \to \beta' X$$ is the identity map.
Putting the last two conclusions together, $$F : \beta X \to \beta' X$$ is a homeomorphism whose inverse is $$F'$$. And $$F$$ is unique (another application of the uniqueness portion of the universal property).
• The restriction to $X$ is the identity remark is only relevant in the completely regular case where $\beta X$ is an "honest" compactification (i.e. $X$ can be seen as a dense subspace of $\beta X$), not so much in also current case where $\beta X$ is always (all $X$) defined as a compact Hausdorff space with a universal property. A reflection into $\textsf{CompHaus}$ (IIRC?) Jan 2, 2022 at 0:37