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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

It is clear that is not homeomorphic to a sphere (by which I mean the surface only, not the interior). The latter is compact whereas the former isnt. However if you remove a single point from the sphere, then the two spaces are homeomorphic.*t* varies from 0 to 2 (including 0 but not 2).

where , we have a homeomorphism!

Let us take the unit sphere centred at

, given by , and let us remove the north pole . Call this pointless sphere . Then is the disjoint union of circles formed by the intersection of with the plane asNow consider

itself. This is the disjoint union of origin-centred circles of all possible non-negative radii (counting as a circle of radius 0). Let be any continuous bijection (e.g. or ).Then if we define

by and for[align=center]

[/align]where , we have a homeomorphism!

I was thinking about this last night. Thinking about math problems is a great way to pass the time when youre having insomnia.

*Last edited by JaneFairfax (2010-10-29 01:50:05)*

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**scientia****Member**- Registered: 2009-11-13
- Posts: 224

The surface of the sphere is homeomorphic to the extended complex plane .

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**coprime****Member**- Registered: 2011-01-15
- Posts: 1

since homeomorphism is defined by continuity but does not require smoothness we can map the punctured sphere to sharpened pencil shape: a truncated half-cone with apex at the south pole intersecting the sphere at the equator, glued to a half infinite cylinder. then the cylinder is easily mapped to the remaining (infinite) portion of the cone. finally flatten out the cone.

use the unit sphere centred at the origin:

for the map of lower hemisphere to truncated half-cone we send

(x,y,z) to (px,py,z) where p=√ (1-z²)

for the map of the upper punctuated hemisphere to the half- cylinder we send

(x,y,z) to (x/r,y/r,zq/r) where r=√ (x²+y²) and q=√(1-r²)

to map the half-cyclinder to the remainder of the half-cone

send (x,y,z) to (-ipx,-ipy,z) where i=√(-1)

to flatten the half-cone send (x,y,z) to (x,y)

(NB the use of imaginaries is merely a notational convenience)

*Last edited by coprime (2011-01-15 08:14:11)*

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