Let $X$ be the orbit space $\mathbb{R}^n /\mathbb{R}^+$ where the action of $\mathbb{R}^+$ is given by $$t \cdot (x^1,\dots,x^n)=(tx^1, \dots,tx^n).$$ Show that $X$ has an open subset homeomorphic to $\mathbb{S}^{n-1}$.
My line of thought was the following. We restrict the quotient map $\pi :\mathbb{R}^n \to \mathbb{R}^n /\mathbb{R}^+$ to a map $$\pi':\mathbb{R}^n \setminus \{0\} \to \pi'(\mathbb{R}^n \setminus \{0\})$$ which is again a quotient map and define $$f:\mathbb{R}^n \setminus \{0\} \to \mathbb{S}^{n-1}$$ by $x \mapsto x/\|x\|$. These are both quotient maps and now I would want to have that they agree on fibers i.e. $$\pi'(x)=\pi'(y) \iff f(x)=f(y)$$ so that I could use the uniqueness of the quotient to conclude that there exists a homeomorphism $$\varphi : \pi'(\mathbb{R}^n \setminus \{0\}) \to \mathbb{S}^{n-1}$$ which would give the desired result as $\pi'$ is open.
I do not know how to verify these assumptions for the maps. Can anyone help me with this?