Consider the 3-sphere $S^3$ viewed as a subspace of $\mathbb{C}^2$: $$S^3=\{(u,v):u,v\in\mathbb{C}, |u|^2+|v|^2=1\}.$$ Inside the sphere we consider $$A:=\{(u,v)\in S^3: |v|=\frac{\sqrt{2}}{2}\}.$$ I already showed that $S^3\setminus A$ has two connected components, namely $$X_1=\{(u,v):|u|<|v|\}, X_2=\{ (u,v): |u|>|v|\}.$$ I also showed that the two connected components satisfy: $$\bar{X}_1 \cap \bar{X}_2 = \delta(X_1) = \delta (X_2) = A.$$ with $\delta$ the boundary. all these closures and boundaries are inside the space $S^3$.
Now consider the unit circle and the closed unit disk $$S^1=\{(\alpha,\beta)\in\mathbb{R}^2: \alpha^2 + \beta^2 = 1\}, D^2=\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 1\}.$$ By a solid torus we mean any space homeomorphic to $S^1\times D^2$.
I have to show that $$f: S^1\times D^2 \rightarrow \mathbb{R}^3, f((\alpha,\beta),(x,y))=((2-x)\alpha,(2-x)\beta,y)$$ is an embedding. Then I have to show that $\bar{X}_i$ is a solid torus for $i\in\{1,2\}$.
How should I do this?
Thanks!