We have constructed a correspondence \(X \to \beta(X)\) that assigns, to each completely regular space, its Stone-Čech compactification. Now let us assign, to each continuous map \(f: X \to Y\) of completely regular spaces, the unique continuous map \(\beta(f): \beta(X) \to \beta(Y)\) that extends the map \(i \circ f\), where \(i: Y \to \beta(Y)\) is the inclusion map. Verify the following:
If \(\mathsf{l}_X: X \to X\) is the identity map of \(X\), then \(\beta(\mathsf{l}_X)\) is the identity map of \(\beta(X)\).
If \(f: X \to Y\) and \(g: Y \to Z\), then \(\beta(g \circ f) = \beta(g) \circ \beta(f)\).
These properties tell us that the correspondence we have constructed is what is called a functor; it is a functor from the “category” of completely regular spaces and continuous maps of such spaces, to the “category” of compact Hausdorff spaces and continuous maps of such spaces.
Solution
Let \(i_X:X\hookrightarrow\beta X\) denote the canonical embedding.
Both \(\beta(\mathsf l_X)\) and \(\mathsf l_{\beta X}\) are continuous maps \(\beta X\to\beta X\) whose restriction to the dense subspace \(X\) is \(i_X\). By uniqueness in the Stone–Čech extension property, \[ \beta(\mathsf l_X)=\mathsf l_{\beta X}. \]
Let \(f:X\to Y\) and \(g:Y\to Z\). The maps \[ \beta(g\circ f),\qquad \beta(g)\circ\beta(f):\beta X\longrightarrow\beta Z \] are continuous. For \(x\in X\), \[ (\beta(g)\circ\beta(f))(i_X(x)) =\beta(g)(i_Y(f(x))) =i_Z(g(f(x))), \] which is also the value of \(\beta(g\circ f)\) at \(i_X(x)\). Since \(X\) is dense in \(\beta X\) and \(\beta Z\) is Hausdorff, two continuous maps \(\beta X\to\beta Z\) agreeing on \(X\) agree everywhere. Therefore \[ \beta(g\circ f)=\beta(g)\circ\beta(f). \] Thus \(\beta\) preserves identities and composition, so it is a functor.