Let \(Y\) be an arbitrary compactification of \(X\); let \(\beta(X)\) be the Stone-Čech compactification. Show there is a continuous surjective closed map \(g: \beta(X) \to Y\) that equals the identity on \(X\).
[This exercise makes precise what we mean by saying that \(\beta(X)\) is the “maximal” compactification of \(X\). It shows that every compactification of \(X\) is equivalent to a quotient space of \(\beta(X)\).]
Solution
1. Let \(i : X \to Y\) be the inclusion of \(X\) into its compactification \(Y\) (a continuous map into a compact Hausdorff space).
A compactification \(Y\) of \(X\) is a compact Hausdorff space containing \(X\) as a dense subspace, so \(i\) is continuous.
2. By the universal property of the Stone–Čech compactification, \(i\) extends uniquely to a continuous map \(g : \beta(X) \to Y\).
Every continuous map from \(X\) to a compact Hausdorff space extends uniquely to a continuous map on \(\beta(X)\); step 1 supplies such a map.
3. \(g\) equals the identity on \(X\).
By step 2, \(g\) extends the inclusion \(i\).
4. \(g\) is surjective.
The image \(g(\beta(X))\) is compact because \(\beta(X)\) is compact and \(g\) is continuous, so it is closed in the Hausdorff space \(Y\). By step 3 it contains \(X\), which is dense in \(Y\). Hence \(g(\beta(X)) = Y\).
5. \(g\) is closed.
A closed subset of the compact space \(\beta(X)\) is compact, its image under \(g\) is compact, and a compact subset of the Hausdorff space \(Y\) is closed.
6. Q.E.D.
Steps 2 through 5 show that \(g : \beta(X) \to Y\) is a continuous surjective closed map equal to the identity on \(X\).