Exercise 2, § 9

(a) In a space $\mathbf{R}^n$ ($n \geq 2$), let U be a non-empty open set with non-empty complement; show that the frontier of U contains a non-empty perfect set (cf. General Topology, Chapter I, § 9, Exercise 17).

(b) Deduce from (a) that, if A is an everywhere dense subset of $\mathbf{R}^n$ which meets every perfect set in $\mathbf{R}^n$, A is convex.

(c) Show that there exists in $\mathbf{C}$ an everywhere dense subfield K which is connected and locally connected and is a pure transcendental extension of $\mathbf{Q}$ (apply General Topology, Chapter IX, § 5, Exercise 18 (b) and (c), constructing K by transfinite induction, using (b) and the method described in Set Theory, Chapter III, § 6, Exercise 24). Deduce that there exists a subfield $K' \supseteq K$ of $\mathbf{C}$ which is (algebraically) isomorphic to $\mathbf{R}$, connected and locally connected.

(d) Show, using (c), that there exists on $\mathbf{C}$ the topology of a connected and locally connected space, which is compatible with the field structure and for which the completion of $\mathbf{C}$ is an algebra over $\mathbf{C}$ which is the direct composition of two fields isomorphic to $\mathbf{C}$.