Exercise 2, § 2

Soit S is the graph of the function from $]0,1]$ into $\mathbf{R}$ given by $x\mapsto$ sin($\pi /x$); let $\overline{S}$ be its closure in $\mathbf{R}^2$.

(a) Prove that the spaces S and $\overline{S}$ are connected.

(b) Prove that a connected and compact subset of $\overline{S}$ containing the points $(1,0)$ and the points $(0,1)$ is necessarily equal to S, but that there is no homeomorphism of $\mathbf{I}$ onto $\overline{S}$ taking 0 to $(1,0)$ and 1 to $(0,1)$.

(c) Determine the arcwise connected components of the space $\overline{S}$.

(d) Prove that the space $\overline{S}$ is not locally connected.