Non-standard topology on figure-eight

Question

In "Analysis and Algebra on Differentiable Manifolds" (Gadea, Masqué, and Myktyuk), Problem 1.35 (page 21), the figure eight $E$ is defined as $E = \{(\sin 2t, \sin t) : t \in \mathbb{R}\}$ and it is then claimed that the map $\varphi : E \to \mathbb{R}$ given by $\varphi(\sin 2t, \sin t) = t$ is, for $0 < t < 2 \pi$, an injection.

Surely that's wrong: both $t = 0$ and $t = \pi$ give the same point on $E$.

1. What might be meant?
2. The problem goes on to say to use the "topology inherited from [this] injection in[to] $\mathbb{R}$." What might be meant by that.

Answer 1

As noted in the comments, $t=0$ is not in the interval $0 < t < 2\pi$.

As regards

the topology inherited from the injection $\varphi: E\to\mathbb{R}$

the author means the topology on $E$ such that a subset $U$ of $E$ is open if and only if $\varphi(U)$ is open in $\mathbb{R}$.