Exercise 13, § 5

Let $X$ be a totally non-meagre space which is connected and locally connected. Show that $X$ cannot be the union of an infinite sequence $(F_n)$ of non-empty mutually disjoint closed sets. (Let $H$ be the union of the frontiers $H_n$ of the $F_n$ in $X$; show that $H$ is closed in $X$ and that each set $H_n$ is nowhere dense in $H$; to establish the latter point, consider a fundamental system of connected neighbourhoods of a point of $H_n$, and argue by contradiction, using Chapter I, § 11, Proposition 3.)