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)

]]>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.

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