Chapter 4

If \(\left\\{X_{\alpha}\right\\}_{\alpha} \in A\) is a family of topological spaces of which infinitely many are noncompact, then every closed compact subset of \(\prod_{\alpha \in A} X_{\alpha}\) is nowhere dense.

Let \(1-\sum_{1}^{\infty} c_{n} t^{n}\) be the Maclaurin series for \((1-t)^{1 / 2}\). a. The series converges absolutely and uniformly on compact subsets of \((-1,1)\), as does the termwise differentiated series \(-\sum_{1}^{\infty} n c_{n} t^{n-1}\). Thus, if \(f(t)=\) \(1-\sum_{1}^{\infty} c_{n} t^{n}\), then \(f^{\prime}(t)=-\sum_{1}^{\infty} n c_{n} t^{n-1}\). b. By explicit calculation, \(f(t)=-2(1-t) f^{\prime}(t)\). from which it follows that \((1-t)^{-1 / 2} f(t)\) is constant. Since \(f(0)=1, f(t)=(1-t)^{1 / 2}\).

Let \(X\) and \(Y\) be compact Hausdorff spaces. The algebra generated by functions of the form \(f(x, y)=g(x) h(y)\), where \(g \in C(X)\) and \(h \in C(Y)\), is dense in \(C(X \times Y)\).

Let \(X\) be a compact Hausdorff space. An ideal in \(C(X, \mathbb{R})\) is a subalgebra \(\mathcal{J}\) of \(C(X, \mathbb{R})\) such that if \(f \in J\) and \(g \in C(X, \mathbb{R})\) then \(f g \in J\). a. If \(J\) is an ideal in \(C(X, \mathbb{R})\), let \(h(\mathcal{J})=\\{x \in X: f(x)=0\) for all \(f \in \mathcal{J}\\} .\) Then \(h(\mathcal{J})\) is a closed subset of \(X\), called the hull of J. b. If \(E \subset X\), let \(k(E)=\\{f \in C(X, \mathbb{R}): f(x)=0\) for all \(x \in E\\}\). Then \(k(E)\) is a closed ideal in \(C(X, \mathbb{R})\), called the kernel of \(E\). c. If \(E \subset X\), then \(h(k(E))=\bar{E}\). d. If \(J\) is an ideal in \(C(X, R)\), then \(k(h(\mathcal{J}))=J\). ( Hint: \(k(h(J))\) may be identified with a subalgebra of \(C_{0}(U, \mathbb{R})\) where \(\left.U=X \backslash h(J) .\right)\) e. The closed subsets of \(X\) are in one-to-one correspondence with the closed ideals of \(C(X, \mathbb{R})\).

Consider N (with the discrete topology) as a subset of its Stone-Cech compactification \(\beta N\). a. If \(A\) and \(B\) are disjoint subsets of N, their closures in \(\beta N\) are disjoint. (Hint: \(\left.X_{A} \in C(\mathrm{~N}, I) .\right)\) b. No sequence in N converges in \(\beta N\) unless it is eventually constant (so \(\beta N\) is emphatically not sequentially compact).

If \(X\) is normal and second countable, there is a countable family \(\mathcal{F} \subset C(X, I)\) that separates points and closed sets. (Let \(\mathcal{B}\) be a countable base for the topology. Consider the set of pairs \((U, V) \in \mathcal{B} \times \mathcal{B}\) such that \(\bar{U} \subset V\), and use Urysohn's lemma.)

Let \(\left\\{\left(X_{n}, \rho_{n}\right)\right\\}_{1}^{\infty}\) be a countable family of metric spaces whose metrics take values in \([0,1]\). (The latter restriction can always be satisfied; see Exercise \(56 \mathrm{~b}\).) Let \(X=\prod_{1}^{\infty} X_{n .}\) If \(x, y \in X\), say \(x=\left(x_{1}, x_{2}, \ldots\right)\) and \(y=\left(y_{1}, y_{2}, \ldots\right)\), define \(\rho(x, y)=\sum_{1}^{\infty} 2^{-n} \rho_{n}\left(x_{3}, y_{n}\right) .\) Then \(\rho\) is a metric that defines the product topology on \(X\).