Sphere vs Plane in Point-Set Topology

[Alan Grayson]

Both are connected. Both have no boundary. Both are closed, since both contain their accumulation points. Both have uncountable elements. So how can they be distinguished within the context of point-set topology? TIA, AG

[Brent Meeker]

A closed curve on a sphere with a point not on the curve can be contracted to a point without crossing the point not the curve no matter where that point is.

Brent

On 1/24/2020 5:38 PM, Alan Grayson wrote:

Both are connected. Both have no boundary. Both are closed, since both contain their accumulation points. Both have uncountable elements. So how can they be distinguished within the context of point-set topology? TIA, AG

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[Alan Grayson]

On Friday, January 24, 2020 at 6:38:43 PM UTC-7, Alan Grayson wrote:

Both are connected. Both have no boundary. Both are closed, since both contain their accumulation points. Both have uncountable elements. So how can they be distinguished within the context of point-set topology? TIA, AG

One way to distinguish the two surfaces is the fact that on a sphere, the path starting at any point, closes on itself, unlike the starting point on a plane. But what if the sphere is distorted, suppose it looks like potato. For a potato-shaped "sphere", this distinguishing feature fails. So the question remains. And the answer is, what? TIA, AG

[Alan Grayson]

On Friday, January 24, 2020 at 6:58:29 PM UTC-7, Brent wrote:

A closed curve on a sphere with a point not on the curve can be contracted to a point without crossing the point not the curve no matter where that point is.

Brent

Doesn't seem right. If we have a circle on the sphere, and a point at its center, your claim will fail. AG

 

On 1/24/2020 5:38 PM, Alan Grayson wrote:

Both are connected. Both have no boundary. Both are closed, since both contain their accumulation points. Both have uncountable elements. So how can they be distinguished within the context of point-set topology? TIA, AG

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[Brent Meeker]

On 1/24/2020 7:01 PM, Alan Grayson wrote:

On Friday, January 24, 2020 at 6:58:29 PM UTC-7, Brent wrote:

A closed curve on a sphere with a point not on the curve can be contracted to a point without crossing the point not the curve no matter where that point is.

Brent

Doesn't seem right. If we have a circle on the sphere, and a point at its center, your claim will fail. AG

No it won't.  Just "expand" the circle till it contracts to the anti-podal point.

Brent

[Alan Grayson]

OK. Is there a formal name for this property? AG 

[Brent Meeker]

Spherical topology.

Brent

[Lawrence Crowell]

You have to go beyond point-set topology and consider homology or homotopy theory, With a sphere a curve is a loop that is contractible to a point. A line in flat spacetime is not contractible. This might make one thinkthere is a homology, or cohomology, of H^1(R^2) = ker(M)/im(M) for M a map. The homotopy π_1 for the sphere is zero, contractible, but is Z for the Euclidean space. One might think the homology H_1(R^2) is the same, but Euclidean plane and 2-sphere have a trick up their sleeve with the stereographic projection so the pole of S^2 gets mapped to "infinity" So the middle homology group or ring is zero. The homotopy fundamental form  π_1 has commutators that make it not zero. However, that stereographic projections means the point at the pole is mapped away so while the sphere has H_0(S) = H_2(S) = Z the Euclidean plane has H_0(R^2) = 0. 

LC

[Alan Grayson]

On Saturday, January 25, 2020 at 5:52:05 PM UTC-7, Lawrence Crowell wrote:

You have to go beyond point-set topology and consider homology or homotopy theory, With a sphere a curve is a loop that is contractible to a point. A line in flat spacetime is not contractible. This might make one thinkthere is a homology, or cohomology, of H^1(R^2) = ker(M)/im(M) for M a map. The homotopy π_1 for the sphere is zero, contractible, but is Z for the Euclidean space. One might think the homology H_1(R^2) is the same, but Euclidean plane and 2-sphere have a trick up their sleeve with the stereographic projection so the pole of S^2 gets mapped to "infinity" So the middle homology group or ring is zero. The homotopy fundamental form  π_1 has commutators that make it not zero. However, that stereographic projections means the point at the pole is mapped away so while the sphere has H_0(S) = H_2(S) = Z the Euclidean plane has H_0(R^2) = 0. 

LC

Why do cosmologists say a hyper-spherical universe is closed, whereas a plane is open, when in point-set topology they're both closed (both contain their accumulation points)? What do open and closed mean? AG