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I am studying the S G Krantz's book, where in the quotient topologies section, while defining the "the space obtained from $X$ by collapsing the subset $E$ to a point"(it is nothing but, taking a subset of the original topological space and obtain a space by identifying each point in the subset to one point and make each points not in the subset to singletons ). Its definition is clear to me. But while dealing with an examples(which I mention as following ) of it in the book, I have some problem to visualize.

Let $X$ be the closed unit ball in $\mathbb{R}^n$ and $\mathbb{S}^{n-1}$ be the unit sphere in $\mathbb{R}^n$. Then $\mathbb{S}^{n-1}$ is the boundary of $X$. If we take the space obtained from X by collapsing its boundary to a point, we obtain a space that is homeomorphic to $\mathbb{S}^{n}$ (the unit sphere in $\mathbb{R}^{n+1}$). Please someone help me to understand this example.

amWhy
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SJA
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  • Not sure if this helps, but for $n=2$ we do it in cartography: https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection . The whole edge of the circular map is "collapsed" to one point - the South (or North) Pole. –  Aug 17 '20 at 18:13
  • Have you studied the universal property for quotient maps? – Lee Mosher Aug 17 '20 at 18:19

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If you connect the ends of a line (closed 1d ball) you get a circle.

If you deform a closed disc (closed 2d ball) to a "bowl" and then contract only the edge of the bowl until the entire edge is contracted into a single point, you get a 2d sphere.

Taking a closed 3d ball... well, your geometric imagination won't do you any good here, and you'll need to do the math, which is probably covered in your textbook. Otherwise, the link in the comments (https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection) by Stinking Bishop will provide you with an idea for a homeomorphism you can try to generalize to higher dimensions.

Vercassivelaunos
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