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 as t varies from 0 to 2 (including 0 but not 2).
Now consideritself. 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 defineby and for
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)
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?
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)