Use two different argumentations to show that $$ W:=\left\{(x_1,x_2,x_3)\in\mathbb{R}^3:-1<x_i<1, i=1,2,3\right\} $$ is a 3-dim. submanifold of $\mathbb{R}^3$.
1) Parametrization (map)
To show that $W$ is a 3-dim. submanifold of $\mathbb{R}^3$, consider the inclusion $$ \varphi\colon W\to\mathbb{R}^3, x\longmapsto x. $$ This is an immersion which maps $W$ to $W$ as a homeomorphism.
2) Don't know a special name for this.
Consider $$ E_k:=\left\{(x_1,...,x_n)\in\mathbb{R}^n: x_{k+1}=...=x_n=0\right\} $$ with $k=n=3$ here. Furthermore consider $$ \theta\colon W\to W, x\longmapsto x. $$ This is an invertible function which fullfills, because of $E_3=\mathbb{R}^3$, $$ \theta(W\cap W)=\theta(W)=W=E_3\cap W=W. $$
So in 1) and 2) I used two different characterizations of submanifolds which I found in my analysis book in order two explain the claim. Would be nice to hear if my both explanations are correct.
With kind regards
math12