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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?

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    $\begingroup$ 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! $\endgroup$
    – user700480
    Jan 1, 2022 at 18:51
  • $\begingroup$ @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. $\endgroup$ Jan 1, 2022 at 19:32
  • $\begingroup$ Do try to chase diagrams, though... $\endgroup$
    – user700480
    Jan 1, 2022 at 19:34

1 Answer 1

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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).

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    $\begingroup$ 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?) $\endgroup$ Jan 2, 2022 at 0:37

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