Topology of punctured plane vs topology of circle?

[nonequilibrium]

So how does the topology of R^n minus the origin relate to that of the (n-1)-dimensional sphere?

I would think the topology of the former is equivalent to that of an (n-1)-dimensional sphere with finite thickness, and open edges. But I suppose that is as close as one can get to the (n-1)-dimensional sphere itself? Hm, they seem pretty similar, but not the same... Is there a way in which they are approximately the same topology-wise?

 

[micromass]

Check out homotopy equivalence and deformation retractions.

 

[Bacle2]

Yes; a homotopy induces an isomorphism of fundamental groups and homology groups:

by functoriality of induced maps , if fg=Id_X and gf=Id_Y , then

(fg)_* =f_*g_* =Id_X* , and

same idea for (gf)_*. Then the induced maps are isomorphisms.

Still, note that the cutsets of the two are different, so that they are not homeomorphic

(tho maybe the fact that the standard map from R^2\{0| is infinite-to-one may give a clue

of this..

 

[nonequilibrium]

Hey, thank you both. Micromass' suggestion made me look up stuff that indirectly made me understand Bacle's post (and more). The homotopy equivalence of spaces is what I was looking for :)

 

[mathwonk]

one has the same topology as the product of the other with the real line.