Exercise 14, § 11

a) Let $X$ be a locally compact and locally connected space, and let $A$ be a closed subset of $X$. Let $A^*$ be the union of $A$ and the components of $CA$ which are relatively compact. $A$ is said to be full if $A^* = A$. Show that $A^*$ is closed and full.

b) Suppose in addition that $X$ is connected. Show that if $A$ is a compact subset of $X$, then $A^*$ is compact (consider a compact neighbourhood $V$ of $A$ and the components of $CA$ which meet the frontier of $V$).

c) Suppose in addition that $X$ is connected and $\sigma$-compact. Show that there exists an increasing sequence $(K_n)_{n \geq 0}$ of full connected compact sets whose union is $X$ and which are such that $K_n \subset K_{n+1}$ for all $n$.

d) Show that the conclusion of b) is no longer valid if we suppose only that $X$ is connected and locally compact, or that $X$ is Hausdorff, connected and locally connected. [Consider the subspace of $\mathbf{R}^2$ consisting of points $(x, y)$ such that either $y = 0$ and $0 \leq x \leq 1$, or else $y > 0$ and $x = 1/n$ where $n$ is an integer $> 0$].