Munkres' Unit Circle Example - Topology on Domain

Question

Consider the $S^1 = \{ x \times y | x^2 + y^2=1 \}$ as a subspace of $\mathbb{R}^2$. Let $F:[0,1) \rightarrow S^1$ be defined by $t \mapsto (\cos(2\pi t),\sin(2\pi t))$. Will the inverse image of an open set containing $(1,0)\in \mathbb{R}^2$ be open in $[0,1)$? I think this will be true regardless of whether I think of $[0,1)$ as the entire space with the usual topology on $\mathbb{R}$ or as a subspace topology on $\mathbb{R}$. I just want to make sure this is correct, because he doesn't specify what the topology should be on $[0,1)$.

Answer 1

Writing $[0,1)$ without further comments always means that it has the subspace topology inherited from $\mathbb R$. This agrees with metric topology on $[0,1)$ induced by the metric $d(x,y) = \lvert x - y \rvert$. Note that this metric is the restriction of the standard metric on $\mathbb R$.

Now you map $F$ is continuous because the two coordinate functions $\cos(2\pi t)$ and $\sin(2\pi t)$ are continuous. Hence preimages of open sets are open.