Is Stone-Čech compactification the only one with universal property

compactificationcompactnessgeneral-topologyuniversal-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?

Best Answer

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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