Homeomorphism between quotient of $R^{n+1 }-{0 } $ and $ S^n \sim $ I want to find an homeomorphism between the quotient space of $R^{n+1 }-{0 } $ and the quotient of $ S^n\sim $. The equivalence relation on  $R^{n+1 }-{0 } $ is defined by $x\sim y$ iff $y=tx $ for some nonzero real number $ x$. The quotient on $S^n $ is defined by the equivalence realtion $x \sim y $ iff $x=\pm y $ ,$x,y \ \in S^n $ 
I tink the following map may work, since it maps points in in the quotient of $R^{n+1 }-{0 } $ to the quotient of $S^n $, (more specifically it maps each line through the origin to one point of $S^n $ ?)
$$f(x)=\frac {x }{||x||} 
$$
Now first of all: is this map continuous? If I take a subset $U $ of $S^n\sim $ then $f^{-1 }(U) $ is a union of lines, hence open sets in the quotient topology of $R^{n+1 }-{0 } $, by the question above?
Secondly I need an continuous inverse than maps points of the unit sphere to lines in the quotient of $R^{n+1 }-{0 } $?
An other edit, it should be an homeomorphism between, the quotient of $R^{n+1 }-{0 } $ and the quotient on $S^n $ defined by the equivalence realtion $x \sim y $ iff $x=\pm y $ ,$x,y \ \in S^n $, not just $S^n $.
 A: These spaces are not homeomorphic: One is compact, the other is not. The correct claim is that $S^n$ is homeomorphic to the quotient of $R^{n+1}\setminus 0$ by the equivalence relation
$$
x\sim tx, t>0. 
$$
Edit: Now I see that you are considering a different equivalence relation
$$
x\sim tx, x\in R^{n+1}\setminus 0, t\in R^\times=R\setminus 0.
$$
Then the quotient is the real-projective space $RP^n$, it is not homeomorphic to $S^n$ unless $n=1$. Take a look at, say, Hatcher's "Algebraic Topology" to see why these spaces are not homeomorphic ($RP^n$ is not simply-connected while $S^n$ is, for $n\ge 2$). 
Edit 2. Now that the question is about two descriptions of $RP^n$, I will give an answer as a sequence of steps you can do in order to solve this problem yourself:


*

*Show that $X=R^{n+1}\setminus 0$ is homeomorphic to $S^n\times R_+$. Hint: the first component of the map from $X$ to the product is
$$
f(x)=x/||x||.
$$
Wha should be the second component of this map?

*Suppose that $X$ is a topological space homeomorphic to the product $Y\times Z$. Consider the projection $p$ of $X$ to the first factor. Show that the quotient topology induced on $Y$ via the map $p$ is the same as the original topology of $Y$. Now apply this to the case when $X$ is as in Part 1 and $Y=S^n$.

*Suppose that $X$ is a topological space and you have subjective maps
$$
p: X\to Y, q: Y\to V, r=q\circ p.$$
Consider the quotient topology on $Y$ induced by $p$ And then the quotient topology $t$ induced on $V$ by $q$. Consider also the quotient topology $t'$ on $V$ induced by the map $r$. Show that the topologies $t, t'$ are the same.

*Now, apply step 3 to the maps
$$
p: X=R^{n+1}\setminus 0\to Y=S^n, q: Y\to V=S^n/\sim,
$$
where $x\sim y$ if and only if $y=\pm x$ and
$$
r: X\to V
$$
which is the quotient map for the equivalence relation
$$
x\sim t x, t\in R^\times,
$$
the group of nonzero real numbers.
