Explain how $\mathbb{P}^n$ can be seen as a quotient of $\mathbb{R}^{n+1}\setminus\{0\}$ modulo an equivalence that is to be specified.

Question

For my introductionary class in topology I have to do the following problem:

Explain how $\mathbb{P}^n$ can be seen as a quotient of $\mathbb{R}^{n+1}\setminus\{0\}$ modulo an equivalence relation that you have to specify.

Now I get that $\mathbb{P}^n$ consists of all lines in $\mathbb{R}^{n+1}$ that go through the origin, and that we can define an equivalence relationship between two non-zero vectors as follows:

$x R y$ if and only if there exists some $c\in \mathbb{R}$ such that $x=cy$.

But the problem is that I don't know how to explain this argument formally. I find that I lack some understanding of exactly what a quotient space entails.

Answer 1

Formally elements of the quotient space $X/R$ are equivalence classes of the relation $R$ and the space $X/R$ is endowed with the quotient topology induced by the quotient map $q:X\to X/R$, that is a set $U\subset X/R$ is open in $X/R$ if and only if the set $q^{-1}(U)$ is open in $X$.

For instance, below I provide a quotation from Engelking's "General topology".

In your particular case $X=\Bbb R^{n+1}\setminus\{0\}$, and the equivalence classes are straight lines in $\Bbb R^{n+1}$ that go through the origin $0$ without the origin itself.