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I am trying to learn quotient topology from the book of Munkres where he writes that

Let X be the closed Unit ball $\{x\times y : x^2+y^2 \le 1\}$ in $\mathbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{x\times y\}$ for which $x^2 + y^2 < 1$, along with the set $S^1 = \{x\times y : x^2 + y^2 = 1\}$.One can show that $X^*$ is homeomorphic with the subspace of $\mathbb{R}^3$ called the unit 2-sphere, defined by $S^2= \{(x,y,z) : x^2+y^2+z^2=1\}$

But I can not find the required function from $X^*$ to $S^2$ which is a homemorphism.Can someone help me please.Thanks for your help.

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I cleaned up the TeX usage a bit. –  Michael Hardy Nov 28 '13 at 18:55

1 Answer 1

HINT: Call the point $\langle 0,0,-1\rangle$ the south pole of $S^2$, the point $\langle 0,0,1\rangle$ the north pole, and the circle $\{\langle x,y,0\rangle:x^2+y^2=1\}$ in the $xy$-plane the equator of $S^2$. The lines of latitude on $S^2$ are the intersections of $S^2$ with planes $z=k$ for $-1<k<1$.

Your map from $X^*$ to $S^2$ should take $\{\langle 0,0\rangle\}$ to the south pole. For $0<r<1$ it should take the points of $X^*$ corresponding to the circle $x^2+y^2=r^2$ to the line of latitude corresponding to the plane $z=2r-1$. And it should take the point $S^1$ to the north pole of $S^2$.

Added: This is quite easy to arrange if you work in polar coordinates in $X$ and cylindrical coordinates in $S^2$. I’ve really already told you what the map is: send the point of $X^*$ corresponding to $\langle r,\theta\rangle$ to the point of $S^2$ whose cylindrical coordinates are $\langle\rho,\theta,2r-1\rangle$, where $\rho$ is whatever is required in order for the point to be in $S^2$; since $\rho^2+z^2=1$ for any point $\langle\rho,\theta,z\rangle\in S^2$, this is easy to determine.

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still properly not cleared.will you please write the exact function.May be then it would help me to understand the concept more clearly.anyway thanks a lot to help me sir. –  humdo Nov 28 '13 at 17:34

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