Exercise 9, § 2

Let E be a topological space containing more than one point. Show that there exists a continuous mapping $f$ of $S_1$ into $F = E \times S_1$ such that $f(S_1) \neq F$ and which does not extend to a continuous mapping of $B_2$ into $F$ (use Exercise 8). Deduce that, if $n > 1$, $S_n, \mathbf{R}^n$ and $B_n$ are not homeomorphic to any space of the form $E \times S_1$, where E is any topological space (see Exercise 3). In particular, if $n > 1$, $S_n$ is not homeomorphic to $S_1^n$, and for all $n$, $B_n$ is not homeomorphic to $S_1^n$.

Show likewise that $\mathbf{R}^2$ is not homeomorphic to the complement of a point in $\mathbf{R}^2$, and that $S_2$ is not homeomorphic to $B_2$ (if false, $\mathbf{R}^2$ would be homeomorphic to the product of $S_1$ and the interval ]0, 1]).