The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

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 →