Example 9.1:

(i)

$S^1 \times \{1\}$ is a retract of $S^1\times S^1$. A retraction is given by $r(z, w) = (z, 1)$.

(ii)

$(S^1\times\{1\})\cup (\{1\}\times S^1)$ is not a retract of $S^1\times S^1$ as we shall see later.

(iii)

$S^1$ is a retract of $\mathbb{R}^2 -\{(0, 0) \}$ and the retraction is given by the map ${\bf x} \mapsto {\bf x}/\Vert{\bf x}\Vert.$

(iv)

Suppose $A$ is a retract of $X$ then every continuous map $f:A\longrightarrow Y$ extends continuously to a map ${\tilde f}:X\longrightarrow Y$.

We shall show later (lectures 12-13) that $\pi_1(S^1, 1)=\mathbb{Z}$ is non-trivial but we present it here as a theorem for immediate use in the next lecture on the Brouwer's fixed point theorem.