= Solution
{c}
The <unit circle> is <Hausdorff> and <second countable>, since both properties pass to <topological subspaces> of the <Euclidean plane>. To exhibit its local <coordinate charts>, consider the four <open> <semicircles>
$$
U_{y,+}=S^1\cap\{y>0\},\quad U_{y,-}=S^1\cap\{y<0\},\quad U_{x,+}=S^1\cap\{x>0\},\quad U_{x,-}=S^1\cap\{x<0\}.
$$
They cover $S^1$. On $U_{y,\pm}$, the <projection map> $(x,y)\mapsto x$ is a <homeomorphism> onto the <open interval> $(-1,1)$, with <continuous> <inverse function>
$$
t\longmapsto(t,\pm\sqrt{1-t^2}).
$$
On $U_{x,\pm}$, use $(x,y)\mapsto y$, with <inverse function> $t\mapsto(\pm\sqrt{1-t^2},t)$. These <coordinate charts> show that \b[$S^1$ is a one-dimensional <topological manifold> with empty <manifold boundary>.]
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