The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that
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the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning
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it’s path-connected: there’s a path from every point to every other point;
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loops contract to points—are null-homotopic;
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the space outside of the horned sphere is not simply connected.
This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.
You would think that, but you would be wrong. Continue reading
