Exercise 10, § 4

a) Let $(X_\alpha, f_{\alpha \beta})$ be an inverse system of topological spaces, and let $X$ be the inverse limit. Let $f_\alpha$ be the canonical mapping $X \to X_\alpha$. For each $\alpha$, let $Y_\alpha$ be a subspace of $X_\alpha$ containing $f_\alpha(X)$, such that the $Y_\alpha$ form an inverse system of subspaces of the $X_\alpha$. Show that $\varprojlim Y_\alpha$ can be canonically identified with $X$.

b) Let $(X'_\alpha, f'_{\alpha \beta})$ be another inverse system of topological spaces with the same directed index set. For each $\alpha$, let $u_\alpha : X_\alpha \to X'_\alpha$ be a continuous mapping such that the $u_\alpha$ form an inverse system of mappings. The $u_\alpha(X_\alpha)$ form an inverse system of subspaces of the $X'_\alpha$; show that if $u = \varprojlim u_\alpha$ and if the $f_\alpha$ are surjective, then $u(X)$ is dense in the space $\varprojlim u_\alpha(X_\alpha)$. Does this result still hold good if the $f_\alpha$ are no longer assumed to be surjective? (Cf. Set Theory, Chapter III, § 1, Exercise 32.)