I need to know if some properties about the topological space $\mathbb{R}^n/{\sim}$ are true, where $\sim$ is a equivalence relation defined by $a\sim b \iff \exists \lambda >0$ such that $a=\lambda b$.
The properties are:
1) Is this space compact?
2) Is the space of rays passing throught a open ball a open set?
3) Is the function from $\psi:\mathbb{S}^{n-1}\rightarrow \mathbb{R}^n/{\sim}$ defined by $\psi(a)=\overrightarrow{0a}$ a continuous function?
4) Is there any relation with the $n$-dimensional proyective space?
(sorry for my english)
Well firstly we can see that $X = \mathbb{R}^n / \sim$ then $X = S^{n-1} \cup 0$ as each ray in the equivalence class of its intersection in $S^{n-1}$ and $0$ is alone in its equivalence class.
1) So is $X$ compact? Well let's show its closed and bounded which is sufficient by Heini Borel. Bounded is easy seeing that every point in $X$ is at most $1$ from $0$. To see that $X$ is closed we consider a function $f: X \rightarrow \mathbb{R}$ defined by $x \mapsto \lVert x \rVert$, the norm of $x$. As $f$ is continous and $f(X) = {0,1} \in \mathbb{R}$ and ${0,1}$ is closed in $\mathbb{R}$ then $X$ is closed. So $X$ is thus compact.
2) The wording on this statement is confusing. I think it false by my reasoning to (1).
3) Yes, cause it is clearly the identity function.
4) Well the $n$-dimensional projective space is $\mathbb{RP}^n = (\mathbb{R}^n-0)/\sim$ where $a \sim b$ if $a = \lambda b$ for $\lambda \in \mathbb{R}$. Its definitely similar but our space $X$ includes $0$ and rays form equivalence classes not lines.