Exercise 16, § 11

Let $X$ be the subspace of the real plane $\mathbf{R}^2$ which is the union of the line $y = 0$, the closed segments with end points $(0, 1)$ and $(n, 1/(1+n))$, and the half-lines $x \leq n, y = 1/(1+n)$ ($n$ an integer $> 0$). Show that $X$ is a connected space but that there is a component $C$ of the complement of $(0, 1)$ in $X$ such that $(0, 1) \notin \overline{C}$ [(compare with Exercises 4 a) and 15 b)].