[Math] Homeomorphism between quotient of $R^{n+1 }-{0 } $ and $ S^n \sim $

general-topology

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 $.

Best Answer

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:

  1. 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?

  2. 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 $Y=S^n$.

  3. 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.

  4. 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.

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