Exercise 8, § 2

Show that there exists no continuous mapping $f$ of $B_2$ onto $S_1$ which extends the identity mapping of $S_1$ (using Exercise 7, show that $f$,

() We shall see later that, for $n > 1$, the result remains true even if $f(S_1) = S_n$. (*) We shall give later a general definition of an inessential mapping of a topological space into an arbitrary topological space, and we shall show there that, for mappings $S$ into $S_1$, this definition is equivalent to that given here.

restricted to the circle with centre o and radius $r \leq 1$, would always be an inessential mapping of this circle into $S_1$, and hence obtain a contradiction when $r = 1$.